RESEARCH ON MARKOFF CHAINS
by
V/l Romanovsky
Institute of Statistics
Mimeo. Series No. 104
Ml.y"
1954
Sweden
•
1936
RECHERCHES SUR LES CHAINES DE MARKOFF
par
VII Rorranovsky
ACTA
Volume
66,
PPo
MATHE~TICA
147 .. 251, 1936, Upsala
Translated by Dan TeichrOt-l, Andrew Boreski, Me Bo Danford, Peter Shadden,
Richard Volk, and Ruth B. Hall under the supervision of Imogene Riddick, Instructor,
and S. T. Ballenger, Associate Professor of Modern Languages at North Carolina
State College of the University of North Carolina, Raleigh, N. C. at the request of
the Department of Experimental Statistics, North Carolina State College. (Edited
jointly by the Translation Service and the Department of Experimental Statistics.)
M9.y
Sweden
17, 1954
1936
~SEARCH
ON MARKOFF CHaINS
In the present article I shall deal with discrete Markoff chains and, in
the first place, with simple chains.
I shall omit the history of the Markoff
chains as well as the exposition of their pvesent nature.
The reader will find
the most interesting exposition of this in a.n article by J. Hadamard and Me Fre'chet
in French (On the discontinuous probabilities of events in a chain) published in
" angewandte Mathematik pnd Mechanik, 13 (1933), 92-91.
Zeitschrift fur
Since the theory of stochastic matrices and of their zeros plays a fundamental
role in the theory of Markoff chains and is intimately connected to the theory
1
of non-negative matrices developed by G. Frobenius
I shall begin my article with
a review of the results of G. Frobenius accompanied with new proofs or indications
of proof of the
mo~t
mgnifinant of these results and of the developments of the
theory of G. Frobenius which are indispensable for the following and which coneern stoehastic
Chapter I.
matri~es.
Non-negative Matrices and
1. Definitions and notations.
Sto~hasti~ Matrio~s.
A square matrix
• •
• • • • • • •
• • ann
where the elements are real numbers, is said to be non-negative i f
1.
I.
G. Frobenius has published three articles on non-negative matrices.
Ueber Matrizen aus positiven Elementen 1. Sitzungsberichte der Akademic
der Wissen~chaften zu Berlin, 1908, 471-416.
2. Ueber Matrizen aus positiven E1ementen II. Ibidem, 1909, 514-518.
3. Ueber Matrizen aus nicht negativen Elementen. Ibidem, 1912, 456-417.
They will be cited below as Fr. 1, Fr. 2 and Fr. 3~
-2l"1t(a,,(~)
We denote it by A or
equalities.
and we write A ~ 0 instead of the preceding in-
lve say that it is positive and we write A> 0 i f the elements a..<f3
are all positive.
A non-negative matrix
is called stochastic if it fulfills two conditions:
o
1.
I)
No line and no colUll11l of ¢ :d.s,;,. empty, that is, does not contain zeros
2.
exclusively.
A stochastic matrix ¢ can be non-negative or positive according to whether
¢~
0 or
¢ ') o.
Let
be a unit matrix, where
c,,(,,(
= 1
(,,(
= I;n) and
c,,(~
=0
(,,(,~ =
r;n; ,,( f
~).
Then, by designating in general a determinant
• • • • • • •
c
by
I c I,
nn
the equations
\AE - At -; A(",)
= o~
IAE
=
-
15 J
';;
15(",)
o.
are called characteristic equations and their roots, the characteristic numbers or
zeros of the matrices A and
2.
¢ repectively~
Fundamental jToperties of Non-negative.Matrices and of Stochastic Matrices.
-3If A> 0, the characteristic equation A(",)
I.
positive, simple, and greater in absolute value
~
=0
has ~ root .£ which is real,
all the other roots of this
equation.
The reader will find a proof of this proposition in Fr.l (p.47l).
is called by
The root r
G. Frobenius the maximal root of A(",) = O. We will call it the maxi-
mal zero of A.
II.
minant
Designating
£l
I).E - A r ~ ~
i f '" ~ r and A
~.,«"') the minor of the element "'c.,(6 - a.,(~ of the deter-
~)",)
)' 0
(~
= l,n)
> O.
(Fr. 1, p.472).
We can say further that the matrix adjoint to the matrix
for '"
~
r i f A., 0, because this adjoint matrix is a matrix whose elements are
If A > 0,
III.
smallest of
~
~
maximal
~
!: of A is contained between
~
largest and
quantities.
a.,(
IV.
A is positive
).E -
=
f
a.,(~
All stochastic matrices
(.,(,~ = I;n')
¢ have !2£. maximal
(Fr. 1, p. 476).
~ "'0
= 1;
~ ~ ~
simple and larger in absolute value than all the other ~ of ¢ i f ¢ is positive;
in this ~ all the minors ¢.,(~ (",) of ¢ (",) ~ positive for '" ~ 1.
This theorem is an immediate consequence of the preceding theorem of G. Frobeniuf
In fact, we have
~
17.,( = .~r;)~
=1
for .,(
= l,n
)
~
then, by III, "'0
=1
is the maximal zero of ¢ if
¢ > O. The other assertions of
theorem IV follow from I and II.
Furthermore this theorem can be demonstrated directly and very simply.
Whether
¢ is positive or non-negative
vJe can write
-4-
. ..... .. . . . .
=
¢(A)
A-I
-t/Jn2
dJ
A- Y'nn
• • •
by adding to the first column all the others, whence we see that
>u
= 1 is a zero
of ¢.
It follows, by starting from the system
A.x~ .. ~ x.( 9'.(~'
(1)
(..(,~
= l,n)
which has a non-zero solution, if A is a zero of
Whence, for all zeros of
Therefore
Let A
indeed a maximal zero of
=1
> O.
o
X
o
13
¢.
The system (1) then has a solution
«(3 .. I,n).
xf3 ., 0
Let us show that, the
obtain
¢,
Aa = 1 is
and ¢
¢ we
are all
of
the
same
sign.
For this purpo Be we de-
duce from the system (1) a consequence which will be very useful in the following.
Let us suppose that
A::r
I AI(cos
Q+i.sin Q) is a zero of ¢.
system (I) will have a solution
Xf3
where
x~
= /Xf3 J
are not all zero, and
(cos Q(3 + i sin Q(3)
'We
(f3 :; I;n),
will obtain from (1)
1AI I xf31
cos (Q + Q(3)
I A/I Xf31
sin (Q + Q(3) ..
:;
~
lX..(/fJ..(f3cos Q..(
..(
-~
.dJ
Ix..( I tp -<8. sin
..(
Q..(,
,
Then the
-5whence
(.,t.,~
(2)
::
I;'n).
This is the consequence which we had in mind.
We deduce from this
For l :: 1 we have Q
=0
and Q.,t.
=0
or n, therefore cos{9.,t. - Q~ - Q) ::
"t 1 and
as in this case
we see that, for ¢
and all the
> 0,
cos{Q.,t. - Q/3) :: 1 (.,t.,p = l,n), therefore all Q.,( :: 0 or n
x~ are of the
same sign.
We can take them all positive and none
among them is not equal to zero because, in the opposite case, all the other
x~ would be it too by virtue of the identities
o
x~
~
0
= .;; x.,t.
which cannot take place beacuse ¢
cp~,
(I) :: O.
Therefore, for l = 1 and ¢ ) 0, the system (1) has a positive solution
o (~
x~
-
= l,n).
This is a solution, unique to within a constant factor, and we
can write
from which we can deduce that ¢.,(~(l) are all different from zero and of the same
sign.
We establish then without difficulty that
(4)
-6from which and from the
Ao ::;
preceding remark we see that ¢'(l)
f
0, that is, that
1 is a simple zero of ¢.
Take now the system
(5)
of which the characteristic equation is
Let A be a root of this equation.
l
(~ '" 2,n)
Then
(5) admits a non-zero solution x(~)
and we will have
(.,(,~
¢ being
supposed positive.
= 2;i1)
As a consequence, all the roots of the equation
¢ll(A) '" 0 are in absolute value less than unity.
This demonstrates that ¢u(A)
is positive in the interval (1, + co).
By
the same procedure we can demonstrate that ¢22(A), ••• , ¢nn(A) and, more
generally, all the principal minors of the different orders that we will designate
by ¢~ .".". .,( I .,(1° ••.,( (A) are positi v~ "
for .
A = 1.
m
m
B-ut then, bY' the
. . remak-made .
above relative to the equations (3), we conclude that all the minors ¢.,(~(h) are
positive for A ~ 1 11
Still we can reason otherwise.
Let us suppose that all the
non-principal minors of peA) and of the orders less than n - 1 are positive for
Then the identitie s of the form
(6)
¢.,(~(A) '" lP.,(~¢.,(~
make us see that
¢.,(8 (A) > 0
\ .(j3(A) +
(.,(,~::;
~
qJ.,('6 fJo ~¢.(j3"'-«36
It,~
'/
(A)
r.;n) for A ~ 1..
It remains only to verify that all the minors of a determinant like
A - to
'( 00
-7are all positive for). ? 1, this we can verify immediately.
Therefore the in-
equalities
(7)
are proved again.
I~reover,
this proof causes us to see the justification of the following
proposition..
v.
All ~ minors
!2.f.
~ the orders of
¢ ().)
~ positive for).? 1, if
¢ > o.
The same reasoning applied to a stochastic matrix which has some elements
equal to zero shows us that all the minors of all the orders of such a matrix
are non-negative.
From this remark and from the equalit.ies
which are easy to establish, we deduce the theorem:
VI.
In order that
Aa
= 1 be
~ ~ of multip.:J,icity
and sufficient that all ~ principal minors of
n-m+l be zero, but at least
~
m of
¢,
~ ~ necessary
¢ ().) of orders n-l, n-2, ... ,
of the principal minors of order n-m be different
from zero.
3.
Decomposable and Indecomposable Hatrices.
We say that a matrix
is decomposable i f it can be placed, by an identical permutation of lines and of
columns, in the form
where P, Q, R, S are the sub matrices, P and S being square
matrices and Q and R
rectangular matrices, and where at least one of the matrices Q and R is equal to
Zero,
-aThe matrix A is indecomposable if it cannot be decomposed in the manner
indicated.
G. Frobenius has established the following propositons on decomposable and
indecomposable matrices which apply almost without modification to stochastic
matrices.
o VII~ If ~ of ~ principal minors of A(r), say P(r), is equal ~ zero,
A(r) ~ decomposable.
~ of ~ principal minors ~ Per) is e9.ual
!f., besides,
to zero, Per) is ~ of ~ indecomposable parts of A(r).
r signifies here, as above, the maximal root of A(r)
(Fr.3,p.459,III).
= O. Let us note
further that P(r) is called the indecomposable part of A(r) if the corresponding
matrix P is indecomposable, P and P(r) being connected by the relation
=} rE
P(r)
o VIII.
If
~
(0) (1)
cannot
~
vanish
(2)
a.,(~ , ••• ,
~ ~ ~ ~
for
~
pl.
~
matrix A is indecomposable
a.,(~, a-'43 '
-
n quantities
(n-l)
a.,(~
choice of
~
indices .,(,
~.
(Fr. 3, p.461, IV)o
We have set here
(0)
a..(~
(1)
a.,(~
()
=
a~)
..(
= 13,
I
~;
= a.J...f3;
~ =
etc., that is
1 for ..(
o for
1 a.,(~
.~
()
--,
a ClB
=
(2) ,
,
1a"(2fat~
k
are the elements of A •
1 By the symbol 0 placed hofore the number of a theorem we indicate that this
theorem, 'Hith the modifications which are always evident, is applicable to the
most general stochastic matrices, that is to say, having any elements whatsoever.
Conversely, in a decomposable matrix we can
~, P
such that
a~~)
~lways
indicate the indices
This follows from the indentity
=
which occurs, if we have the decomposition
A =
oIX.
For r to
~
r: : 1
the maximal
~ ~
A
~
multiplicity k,
!!:.?:!
necessary
and sufficient that k of these indecomposable parts of A('X.) reduce to ~ for
A.
::0
(Fr. 3, p. 461,
r.
V).
Suppose that A may be decomposed as follows:
A=
Pll
0
0
·..
0
P21
P 0
22
• ••
0
·• •.
• • • • •
p
Pm2 P
Pml
m3 • ••
mm
where P..(j3 are the sub matricES of A, PM being the indecompo sable square matrices.
Then the stated theorem means that the necessary and sufficient condition for r
to be the maximal multiple zero of order k (k ~ m) is the disappearance of the
quantities P~(A.) for A.
= r,
setting alw~s
"
IX bis.
¢ it
~
A.a = 1
~ ~ ~ ~
of multiplicity
~
of
~
stochastic matrix
is necessary ~ sufficient ~ ~ should have ~ indecomposable and isolated
diagonal f ields
~
put in the
1
¢
=
!£!!!!
11
121
0
0
• •
·0
1 22
o • •
• 0
• • • • • • • • • • •
Lml
Lm2 L • ••
m3
m ~ k,
-10-
We call the sub-matrices L~~ fields of
fields of
¢ and
¢,
the sub-matrices L~ diagonal
we say that a diagonal field L~ is isolated i f the fields
L~l' L~2' ••• , L~, ~ _ 1,
situated in the same line as
L~,
are all zero.
The sufficiency of the condition stated is shown by the equations
L~.~. (>..)
~
each admitting >"0
=
=0
(i
= r;k)
~
1 as a simple root, l.vhen
L~
..( ,
1 I
L~ ~
2 2
,
u.,
L~ ~
.l( l(
are indecomposable and isolated.
To see the necessity of this we note that, by the preceding theorem,
¢ has
k indecomposable diagonal fields admitting \, = I as a zero.
Therefore, we should have
L,;( ..,{. (I) = 0
~
(i =
L..(.~. being indecomposable.
~
I;k) ,
~
Let us prove that thesefJ.elds are, furthermore,
~
isolated.
L~.
Let us consider one of those fields, designating it by
indecomposable, all the minors of
L~(l)
Since it is
are positive, as results from theorem
XII proved above and independently from IX b.
Suppose that
are not all zero.
- , ,
-18'
Then, by designating the elements of
s
= order of
cannot all be equal to 1.
L~)
L~
we conclude that the sums
Let
II,
;(fO
by
rf
1, therefore <: 1.
1f6
-J1add now to the column
6
of L..<..<.(h.) all the other columns and develop L..<..<.O..)
according to the elements of this column; we shall have
...
where (JJ06
that
are the minors of L..<..<.(h.) of the column
>u '"
1 cannot be a zero of L
M
indecomposable, and 1 -
Jl1so
6. We see from this identity
being
, because't'l)6 (1) are all positive" L
M
is greater than zero, therefore LM(l)
We conclude from this that LM
> O.
must be isolated, since L..<.../1) .. 0 and ~ is
indecomposable: the necessity of our condition is therefore proven.
!,
~
of
greatest of
~
elements aM
If !: is !: multiple maximal ~ of
X.
place; either
~ ~ ~
~
is equal
~ ~
~
alternatives
~ if
~ ~
it
principal minors of order n - 1 of A( h.) vanish for h. '" r
!.! not,
=: principal
(Fr. 3, p. 461, VI).
minor of order n - 2.
We saw from theorem IV that the maximal zero of a stochastic matrix is always
equal to unity.
Therefore, theorem X applied to stochastic matrices can be stated
as follows:
XI.
If
A.a '"
1 ~ ~ ~tiple ~ of
tJ ,
then ~ of ~ elements
fJ.t.,.t.
is
equal to 1 (and ~ all ~ ~ elements of the l~~.t.. ~~) ~ every
principal mino..::. ¢,1.,)1)
(..(
= r;n)
of
¢ (1) vanishes for
~ of the minors
¢.I..\31 ~ (1).
oXIl.
If A ~ indecomposable all ~ minors A-<{3 (h.) ~ positive for h. ~ r.
We deduce froll theorems VII and XII that the neoessary and sufficient condition
for a non-negative matrix A to be indecomposable is that all the minors
should be positive for h.
4.
~
~ (h.)
r.
Decomposable and Indecomposable Stochastic Matrices.
We shall add to the theorems expressed in the preceding sections several
others whi::-h are in the
matrices.
~arne
order of ideas and which concern expecially st.ochafftic
-12..
XIII.
~
p • (: :)
be a decomposition of ~ where R is indecomposable and P positive.
Then for
Ao = 1
to be a multiple zero of ¢ it is necessary and sufficient that Q should be
null.
In this case
>u '" 1 is
a double zero of ¢.
The sufficiency of the condition stated is almost obvious.
matrix, if Q • 0; therefore it has
because P is positive.
~
Inthis case
"0
P is a stochastic
'" 1 for a zero which is then, for P, simple
II
1 is a double ze:ro of ¢ because R being
a stochastic matrix and indecomposable \,- 1 is a simple zero of R and ~ (A)
lit
peA) R(A).
let us prove the necessity of the conditon.
Ao ..
1 is a simple zero of R.
Consequently, it can only be a multiple zero of
¢ under the condition that peA) vanish for A.
P
> 0;
We have already remarked that
= 1, because
¢ (A) • peA) R(A.). But
therefore by (III) the maximal zero of P should be contained between the
greatest and smallest of the sums of the elements taken by lines.
There
necessarily will be among these sums, some sums less than unity if Q contains
non-zero elements and then the maximal zero of P cannot be equal to 1.
Therefore,
for P(l) to be equal to zero it is necessary that P being positive Q should be
equal to zero.
Let us remark here that following the terminology of G. Frobenius, the ma:t:.rix
¢ is said to be completely decomposable if it can be placed in a form such that,
for example,
~
~)
o
¢ .. (:
0
R
where all the fields are equal to zero except the diagonal fields.
-J3.XIV.
>u . . 1 to be !
For
multiple ~ of
¢
~ is necessary ~ sufficient
that the minors
~
(t '" 1, 2, ••• ,
should be zero for every element
tPcI.f3
- 1, ~ + 1, ••• , n)
I- o.
Indeed, let us take ¢~13 ().) and add to any column whatsoever, say if ' (
t t/J),
Then, developing ~~~().) according
of this determinant, all the other columns.
to the elements of this column we shall have
(~
• 1, 2, ••• ,
~
- 1,
13 +
l, ••• ,n)
where
Now let). = 1 be a multiple zero of ¢.
o (~=
f~13
I-
But we already have ¢~~I~~(l) ~ 0 and l-If:
l,n).
¢13 2f l ~~(l) = 0 for all 'If
0, we should have
On the other hand, i f for all
¢~!J ~~(l)
=0
we shall have ¢13~ (1) '" 0
XV.
If
Then, by VI, we should have ¢~I3(l) =
~
(2f= 1,2,
(13'" l,n) and ).
have the decomposition
L
ll
1 21
0
0
•••
0
L22 0
•••
0
..... . . . . .
¢ =
~!! ~ system
~ ~ po~it:1.ve
tp~ /:
solution
"'?'-«3'
therefor~ if
113·
0 we have
, ~ - 1, ~ + 1, ••• , n)
1 will be at least a double zero of ¢.
o
x~
>0
(~
... l,n)
we shall have
L
2l
.. 0,
L)l
..
L)2" 0,
L
...
L
.. ......
ml
and the matrices L.(.,(.(.,(
IS
m2
I,m)
II:
•••
...
L , m_ 1
m
~ ~
= 0,
>u = 1 for maximal
~.
This theorem is a simple consequence of theorem XIII, Fr. 3, p.466.
But we
will prove it here, following the reasoning of G. Frobenius, by introducing
several definitions and procedures which will be useful in the following.
Let, for simplicity
P, R, U being squared sub-matrices of the orders k, 1, m respectively.
The system written above can be put into the condensed form
(8)
X
=
X¢
which is U8ual in the theory of matrices.
Setting
••• ,
we can replace (8) by the system
(9)
Xl ... XlP + X2 Q + X) ~
X2
I:
X2R + X)T, X)
= X)U.
Similarly the adjoint system of (8)J
(10)
xm)
-15-
or
Y=
(11)
¢ Y,
can be replaced by t he system
(12 )
The
Y1
= PYI' Y2
Y
3
= SY1
~Jstem
+
=
QY1 + RY2 ,
TY 2 + UY •
3
(11) has a positive evident solution
o
(13)
y..< ""
(..< ""
1
Let us suppose that (8) has a positive solution Xo:
which we will designate by YO.
x~ >
l,n)
(13 "" l,n).
0,
Then we can write
'\oThence
therefore
o
0
0
0
since the scalar products X2 QYl and X RY I are
3
show that we should have
.
non-negat~ve.
These equalities
Q = 0, S "" 0
o
because X2 .). 0,
0
X > 0
3
0
and Yl > O.
multiplying next the equation X~ = X~R + X~T by Y~ and noting that
0
0 0
Yo
2 = R Y3, we shall obtain X3TY 2 = 0, hence T = O.
By
-16Therefore, Q,S,T are equal to zero.
matrices and A
O
XVI.
=1
Consequently P, R, U are stochastic
is the maximal zero of each of these.
If ¢ is indecomposable, but!:£!: m integral and positive, ¢m is decomposabl
¢m is completely decomposable.
¢ being indecomposable, every minor ¢..(~ (1) is positive (by XII); therefore
Au
1 is a simple zero of
::I
¢ and the system
X
= X,0
admits a positive solution XO.
Then we can v1I'ite
O
X ¢
O
Similarly we shall have X
= XO ¢2
= XO ¢3
or
xO = XO ¢2.
and so on.
O
Therefore X
= xO ¢rn:
the system
admits a positive solution.
We have further
and generally
from which it follows that the matrix ¢m is a stochastic matrix and the system
y
= ¢m Y admits
° = 1 ( = -l,n).
the solution y~
~
Consequently, because of the theorem XV, ¢m is completely decomposable.
We conclude this section by indicating the following important theorem due
to G. Frobenius.
XVII.
o
parts
(~
!
decomposable matrix A is not able
~
fields) of two different types.
One always supposes that /J,. is non-negative.
be decomposed in indecomposable
(Fr. 3, p. 474,(fl12).
-17"
5.
Primitive and Non-primitive 1'1atrices o
We shall say with G. Frobenius
that a non-negative matrix A is primitive if its maximal zero r· is greater in
absolute value than every other zero r' of
A:
Irl
the matrix A is non-primitive i f among the zeros
Likewise, a stochastic matrix
~, different from
A.O
~ such that \~l
:=
:=
> I rtl •
r' f r there
¢ is said to be primiti'le
1, is such that
I'b I'"
1.
On the other hand,
is such\r'{
= Irl.
if all its zero
If there are among these zero s
1, we shall say that ¢ is non-primitive.
Now we shall state the most important theorems on primitive and non-primitive
matrices discovered and proved by G. Frobenius and some new theorems which concern
primitive and non-primitive stochastic matrices in particular.
oIVIII.
If A is ~ non-negative and primitive matrix, ~ of its powers, AP ,
~ ~ positive ~ all the follo1-ring powers
~
powers of A is positive, A is
.!£
OXIX.
~
~
+l,
AP
2
AP , •••• Inversely, if
+
(Fro 3, p.463, IX).
primitive matrix.
non-primitive matrix A all
~
principal clements, a.,,(,,(,
(Fr.
"xx.
order
9!
Every'
~
of
~
Rrimitive matrix is primitive.
A then '. A, 1;.2, ••• , An
~
indecomposable,
~ of
4.
~~.
3, p. 463, X).
fV1d i f when n is
~
is primitive.
(Fr. 3, p. 464, XI).
o
Let
XXI.
rf)
~(A.)
= A.n
+
n'
a l A.
the characteristic function of
n '7 n' 7 nil
••• and a', a" J
~
•••
+
nil
aliA.
+ ••• be
indecomposable non-negative matrix 4, where
~
different from zero.
Let k be
common divisor of the differences n - n', n' - nil, •••. Then A
k
= 1.
power of
~
~
greatest
primitive
.!f.
~ the other hand, A is non-primitive if k. .> 2., and then A~is thelowest
A :which is completely. decomposed into primitive parts Whose number is k.
If one sets
-18-
••• +
+ ••• + a ,
m
~ equation ~(A)
=0
has ! ~ which
!:! positive,
(Fr. 3, p. 468, XIV).
value than all its other roots.
/) XXII.
Suppose
~
simple and greater ~ absolute
non-negative matrix ii, indecomposable and non-primitive whose
lowest power
~ Ak , which ~ completely decomposed ~ indecomposable and primi-
~
Then, by
parts.
it in
~
~
identical permutation of lines
following cyclical f arm where all the parts,
~
~
columns,
~ ~
sub-matrices,
~
place
zero,
(Fr. 3, p. 469,979).
o
o
o
d.
(14)
=
o
• ••
o
• ••
o
• • • • • • • • • • •
0
o
·..
~lO
o
• ••
o
Hereafter we will call L , L , ••• ,
12
23
o
~l
the principal fields of the non-
primitive matrix :.. placed in the form (14) and all other sub-matrices L.,(~, nonprincipal fields.
1ie shall say that the indecomposable and non-primitive matrix ~J. is of the
index k if the lowest power of
which is
l·t.
~,
is decomposed completely in primitive parts
We shall also say that such a matrix is cyclic of index k.
This
designation is justified by the fact that it can be placed in the indicated form
(14) and also by the following considerations.
Let us suppose that .... is indecomposable.
Then..J, can not contain columns or
lines of which all the elements are zero, so that every column and every line contains at loast an element a..(B' which is not zero.
It results that we are always
-19able to find a series of elements
a~l' a A
Q
t'1""2
,
•• "
which are all different from zero, the indices
aa
~
. k-l
~, ~l'
••• ,
~k-l
being all different
vie call such a series a cycle of the order k of the matrix ....
from one another.
and cyclic matrix J.~ of the ~~, if all its cycles are of orders diviSible by
k, k being the greatest conunon divisor of these orders.
It is clear that a~ .. 0 (~
that a.(8
1 0,
II:
!;ii), if
k ~ 2, and that a~~
II
0 every time
if k> 2.
One can now prove the following theorem.
XXIII.
!£!: !!! ~
~ ~
roots
that !. non-negative
£f. ~
~
indecomposable matrix
.~
have
~
equation
(15)
where r is the
~
cyclical matrix
maximal of ...., it is necessary and it suffices
£!. index
~
4 be !
k.
This theorem is demonstrated in my note of thellBulletin de 1a 50ci~t'
l>1athematique de France" LXI (1933) pp.219-219.
I shall reproduce here its proof
because it plays an important role in the following.
We shall start by the proof of the sufficiency of the condition stated.
(16)
The development of the determinant d(A) can be written in the form
n
~(A) .. An + ] (_ l)h An- h ~
•••
h-1
.,( (h)
an.-( ,
n
where (~~2 ••• ~n) - + 1 .. or - 1 according as the permutation ..(.1' .(2 ' ••• , ,,(n
of the numbers 1, 2, .•• ,n consists of an even or uneven number of inversions and
~
designates a sum taken for all the permutations
..(.(h)
~,
..(.2' ••• , ..(.n which
leave fixed n - h of the indices 1, 2, ... , n, these indices being taken in all
c.ho possible combinations and the correspondin,g eleroorzt.s ai i ' a 1 ' ••• ,
i
1 1
';;'i
i
n-h n-h
' 'Hhich entor into the product ~ a 2.(
1
! 2
• •• an-<. ' being replaced by unity.
2
n
Now the matrix A. is cyclical of the index k, then under tht;; sign ,]
..«h)
one
shall encounter only terms composed of cycles having orders divisible by k, from
which it results that the number h must be a multiple of k.
One sees from this
that A(A) can be written in the form
(17)
+ A
•••
where lJ.k
~
~
).,n-lJ.k
,
n.
Now let ~ be any root of (15).
(1 + A. l r
=(i)
~
One will have
-k +
-2k
2r
+ ••• + A
... + A.
n
r-lJ.k )
~
rn-~ )
~
=
A (r)
0,
which proves the sufficiency of our condition.
Let us note that one obtains thus equality (17) b8ginning with the well
known formula
A (A.) =
).,n _
n-l
).,
11
.5!
..(1
n-2
a
..(1..(1
+
A.
..~
.6J
21
,
..(1..(2
a
a
a
a
.,(,1..(1
..(2..(1
..(1.,(,2
- ...
..(2..(2
In order to show the necessity of the condition, let us take one of the primitive roots of the equation (15), let:
Al
=r
( cos
2n
~
+
i sin
2n )
k".
fhen the system Al X = X will have a non-zero solution
(6
= I;ii)
-2Q.-
and one is able to write it, by virtue of the relation (2),
(18)
r
o
0
:::I
)x~1 = flx~1 a~~ cos (Q~ - Q~- k2n
)
This equality shows us that one will have
(19)
if for every
~13 f
0 one supposes cos(Q~ -
Q~
-
~n
f
)
He are going to prove
1.
that this supposition is impossible.
Let
cos(Q~ - Q~
-
~
)
f 1
for one
a~ f O. Then we will have (19). The
conclusion remains true ~ fortiori if cos(Q..( - Q~
-
~n) f
Let us note now that, A being indecomposable, every
1 for every
A~~(r)
a~ f
O.
is positive(by
XII), then the system rY = AI has a positive solution
o >
y~
(..( = !;n).
0
Let us multiply the two sides of (19) by
o
y~
and make the sum for
~
z
r;n:
one
will have the lJ1lpossibili ty
Therefore we ought to have cos(Q~ - Qp
- ~n) = 1
each time that a..(~
f O.
But \-re have noted above that the matrix J\., W1 ich is indecomposable, must have
cycles such that:
where all the elements are different from zero.
2n
cos(9, - Q ---k)
'"
61
Consequently, for one such cycle,
= 1, ••• , cos(QQ
"'m-l
2n)
- Q - --k
~
=1
-22where
a l , a 2, ""
By adding these
am being whole numbers.
equaliti~s,
one obtains:
from Which it results that m must be divisible by k.
In this manner. we see that the orders of all the cycles must be multiples of
k, if 4 allows as zeros the roots of the equation
(I,):
the necessity of our con-
dition is thus proven.
One can see now that a matrix of index k is at the same tire a cyclical
matrix of index k and inversely.
.~
In fact, if
is of the index k, one can write
it in the cyclical form (14), from which one sees immediately that •.a. is cyclical
of the index k.
Inversely, if 4 is cyclical of the index k, its characteristic
equation is nece ssarily of the form U7), thu~ A can be only of the index k and must
be Feducible to the form (14). Thus we are able to conclude:
XXIV. In order that ~ non-negative and indecomposable matrix 4 be cyclical
o
of the index k., it is necessary and sufficient
~ ~
~ ~
should be
place
it in the form (14).
Let us draw somEl consequences from theorem XXIII.
OXXV.
If ,,~(O) :: .i.'~
f
0, ~ matrix
~.
is a divisor of the order of
~, .~ ~
.u
~ ~ cyclical only ~ ~ ~ which
:!:f.
Consequently,
the order n of
either non-cyclical £! cyclical of the index n.
In the latter case, one can place
. ...
~
·..
0
0
a
23 • ••
0
0
0
a
0
a
in the form:
0
0
•
0
.~
12
.• ....·...
0
0
·.. an-l,n
'11
(\
4
is the number
-23nVI.
If
A
is an indecomposable and cyclical matrix of the index k, the
characteristic equation of A can be written as follows:
Y
A(~) = ~
k
(A - r
k
k
)(~
k
k
k
= 0,
- Clr ) ••• (~ - C~r )
where
(y + ~k + k
6.
i'roperties of Minors of Stochastic Ml.trices.
In this number we shall
We call minor ¢~~ of
consider exclusively stochastic matrices and their minors.
the matrix
p the
AA.VII.
matrix that is obtained by omitting line
~
and column a in
¢.
For every stochastic matrix ¢ :
(eo(,~ =
'b
= n)o
being ~ ~ of
P such
~ I~I
f
l,n)
I.
Since ¢(~) = 0, the system ~X = X¢ allows a non~zero solution x~(~ = l,n).
Consequently, if the
¢eo(~(~)
can not have all nuls (then our theorem is
c~rre~~),
one can write:
(20 )
But from the J."d ent'J.es
A.. l
'llXI=l
t'
." x 0 't'.
If)
=~
~ eo( "<15
'
one
'-:'
0
=~x
eo( eo(
' •
e:>nlll ludes.
,
(21)
and (20) and (21) show that the theorem is true.
AxVIII.
Suppose ¢ be
~
indecomposable, non-primitive stochastic matrix ana
of the index k, reduced to the cyclical form:
-24-
(22 )
~
where h
-
h 1 (h
-n, +
0
1u
0
0
0
•••
0
~3
• ••
0
= l,k;
..•
• •
• • • •
Q
0
0
0
·..
1;u
0
0
•••
k + 1 being replaced by 1 in ~k k 1) is a sub-matrix of
-
-
~ lines ~~ ~+1 .££.ll.'ll'~ (t1 + t 2 + ... + t k
-
)
= n);
,+
---
- .._---
= ¢2~(>Uh)
¢lp(tuh)
1
H'
= ~t1+t2H"A(AOh )'
(23)
• • • • • • • • • • • • • • • • • • •
~II
~1 +
¢t
¢t
(24)
•
1
1
•
1
II
R
+~k-1+ 'i"
•••
2
•
•
•
~ (hah)
•
•
•
= ...
~h
•
•
•
•
GI
•
k
.~ ¢.(
~"1
A
i''''
.'t"
¢Ji3 (luh)'
and
(25)
=~... A(A...h);
-U
Aoh~l~ (Ach)'
+ 1,p (tuh) ..
+t +1,
(AOh )
(Ach) .. 0,
if .,(1' ~, ... , .,(k satisfy ~ conditione
•
•
•
-
Consequent.1y, whatever
= ... = ¢t1~(~h)'
+ 1 R(~h) • •••
¢t
-
suppose aged,.!! tuh ~ of ~
roots of the eaua+,:i.cn ).k - 1 • 0 be different from 1.
-
-
..
A~~l ¢1~ (~Oh)'
-25In fact, the system:
AY .. ¢ Y,
(26)
because of (22) is reduced to:
"Ihere
Yl .. (Yl' ••• , ytl ), Y2 .. (ytl + 1, ••• J yfl + t 2 ), ••• , Yk
.. (yel + ... + ek _ l + 1, ' •• J Yn)f.
In order that A. .. Aoh (26) will have a non-null solution:
because, ¢ being indecomposable and cy·clical of the index k,
zero of
¢ (by
IX) and hOh is also zero (by XXVI).
~
.. 1 is a simple
.1
Consequently, ¢(AOh ) ..
~ ¢~",/"oh) being different from zero, the ¢c43 0Uh) are not all zeros and one
wi. 11
..<
have:
This solution yO, for a nearly constant factor, is unique and by (27) one sees
immediately that it can be written as follows:
0
0
Yl .. Y2 ..
(29)
u.
.. Y0t
.. 1,
l
o
0
0
Yt +1 .. y/- +2 .. ••• .. Yt +.8
1
1
l 2
• • •
.•
• • • • • •
.•
. . "Dh'
• • • •
0
0
k-l
Yt + ••• +t +1" ••• = y.n ... AQh •
1
k-l
In fact, these quantities satisfy (27) for A .. AOh •
-26..
From (29) and (28) one sees ~nediately that the equalities (23) and (24)
are permissible.
One will have next:
k
i~Y~
.. 1 + Ach + ... +
~~l =
... 0,
therefore, b7 (28)
k
.~ ¢.,(.
(30)
1"1
if .,(1'
~,
>l
(Ach) .. 0,
1'tJ
... , .,(k are chosen in such manner that
(h ...
!';k).
The equality (22) can be written in the form.
I
lLll
¢ .. i ~
I
1
Lu
•••
~k
~2
•••
~k
\. . • • • • • •
\L
, kl
by setting
~'t
•••
~
= 0 when the sub-matrix LQ-t does not belong to the series of the
principal fields L ,
12
9"c4' (A)
~
• •
~3'
•• ~, Lkl which are not zero. vJe will say that a minor
belongs to the field L~h. if it corresponds to the element ¢~.( which is one
of the elements of the sub-matrix
!g't'
that is, if r.p~.,( C I~h' by using the well
known sign of the theory of uniformitiesQ
Let us envision now the quantities ¢..,t..«A) and ¢043 (}.)
(.,( f t3)
0
One has the
developments~
(31)
= Xn-l
..
-
... ,
-Z7-
rP~ ~~
tP-).-). rfJ~""2 - ...
1J""2""1 rp""2""2
where the sums are taken for all the possible val ues of
"" in the first development and the values"" and
~
""l"
""2 ••• except the valm
in the second.
Let us set for brevity:
(33)
(34)
• • • • • • • •
S(h)
""~
~
=
""1' •••'''1t-1
,
{J~
~~
•••
rp~.l
tf~
f~~
•••
~l""h-l
• • • • • • • •
.. . • • • •
t(J~_1~ tf~-l""l ... ¢J""h-l~-l
and demonstrate the following theorem.
XXIX. The minors ¢M(A) and ¢-«3(A)
("" /;~) of the determinant ¢(A) of ~
stochastic matrix which is indecomposable, non-primitive, of the index
in ~ cyclical
(35)
(36)
¢M(A)
!£!!!!
~
and placed
(22), have ~ developmen~:
= An-
k A.n -k-1
1
+ ( - 1)
n s l
s 1 A- -
¢..«3(A) • ( - 1) -
~s-l)l
kJ
()
s~
2
+ ( - 1) k
S(s) + ( _ 1)s+k-1
""~
An-2k-l
(2k)
I
n s k l
A- - -
( s+k-l)J
(2k)
8M
+ ••• j
S(s+k) +
""~
...
-28.
forrf)-1D. (L..
i
S >0,
~, ~+s,
"'tJ
¢
(X)
4
for rfj
'f .,(,.. C
(.l
~
and
,n-k+s-l
~
(k - s -1)1
( _ l)k-S-l
L..
~,~-s,
(
.,(~
S k-s
)
+
(1)2k-s-l
-
n-2k+s-l
(
\
S 2k-s+
(2k - s -1)1 .,(~
••.
X
s ~ 0.
Let us consider first ¢.,(.,«~).
Evidently ¢~.,(l
=
° (.,(1 = !;n).
The same way,
all the determinants:
are zero i f k /' 2, because ¢ is cyclical of the index k.
• • • •
should be equal to zero i f h
In general a determinant
• • • •
"O(nodk) because otherwise one would have in ¢ cycles
whose orders are not divisible by ko
In this manner by taking into account the
equalities (31) and (33) one verifios that (35) takes place.
Let us consider in the next place ¢.,(~(X), -< ~ ~.
positive or negative whole number or
Suppose that ~~
L1 h --
. . ° and let
zero~
us consider tho determinant:
(l.,(~
f~
• ••
rf.,(l~
tjJ.,(l.,(l
• ••
• • • • • • • •
rf~_l~
Let ¢'"'I'"
I(.lC L. . +
' where
:;.,~ S
0
•
f"'n-l~
~~..l
fll"'n-l
• • • •
...
f "'n-l"'n-l
-29I say that all the members of the development of this determinant are zero
if h
rs
(mod. k).
In order to demonstrate it, let us note that the members of
of the two forms: 1)
rfJ.J..~ 1J""l.J..ll
is a permut.ation of.the
where .('1'
.J..'2' ••• ,
fJ.<2.J..'2
numbers~, ""2'
fJ~-l"<"h-l'
'b
1
where .J.. 1'
are of one
~12J
... , "<'h-;
, ~-l; or 2)fMrlCP""1..<t2."~_1.('h'
"<'h is a permtation of the numbers ~, ""l'
""2' ••• ,
""h-l
in which ..<1 1 ' ~.
But
everyep~¢J""l"<'l ... r(J-'n-l.(I'
h l
= 0,
because
qJ~
= 0, and it remains
to be verified that every member of the second form is equal to - 0, i f h .,. s
(mod k).
Now we c an write:
=
(t{J..< 0
1
~,').
6
•••
rfJlf p-lfj)( ¢J~P
y'
0
p+l
1Jvp+l Qp+2
~ ... tflt:.r-l ~p).. ,~
(l
by transposing the factors and revealing the cyclical groups of factoISsuch that:
anda single group
t(J({1 "0 2
which is
incomplete~
..·tfJ?f.p-l!'"~
cyclical.
By noting that no other factor of M can be different from zero, i f it does
not belong to a principal field of ¢, one sees that the factors of the groups of
11 mU:3t necessarily belong to the consecutive principal fields in -order that M
be
d~fferent
from zero.
Now we must have:
-.3)-
for s ; , 0 and
y
(J
1
C Li . l) If) ,
,J.... /Cf1
y
(J
Co L. 1 . "l
2
J.+, J.+L,
l1J y
c ~" 2
'f ()k-i+l, ?f k-i+2
•••
,tlJy
... ,tfJ
r~p-l~
'r 0k_i, V'0 k-i+l<:
c:::. L
1..
-k,
1
J
1
q- ,q
for s < 0, that is one must have
P
i: S
(mod k) for s
>
0
and
p = k - i + q, q ;; i + s (mod k) for s
< o.
In every case, we must have
(38)
p
-=
s (mod k).
Suppose tJ1at thi. conditon is sat.isfied and let us consider the cyclical
products
su~h
that:
...tflt.r-l 't p
From the fundamental 8uppositon that
¢
•
is cyclical of the index k, it results
iJnmediately that this produtlt i. equal to zero, if
(38 bis)
The congruences
r - p ; 0 (mod k).
(38) and (38 bis) show us obviously that
L\ = 0,
i f h ; s (mcxik).
-31It is easily seen that equalities (J6) and (37) are quite varied when ~~ =0
and (L..
where s is a whole number ~ O.
J.,J.+s,
.......
In order to see it, one has to
write (32) in the form
n-l
.~
¢..(A(A)
=
t-'
~
h=l
and take into account that
,..(h)
-~ II
=- ~
~•
..(1,·· ·~-l
0..(/3
Let us suppose finally that cf~
~~
I:
O.
Then one ought to have either
L i ,i+1'
(
(i
= 1,
k -
1),
or
In every case the member
will be zero, if
h - 1
t
0 (mod k),
and the members are such that
•••
will be zeros, if
h
f
s (mod k),
-32where s
=1
or - (k - 1) according as tfJ~ (Li,i+l, (i :: 1, k -
Then, by (39) one sees that for
epJfJ rf-
or ~~ eLk,l.
0, the coefficients
... (k+l)
o..(~
1),
•••
,
alone are not zeros in general so that we will have in this case
(40)
¢ ..LA (A.) .. A. n-~
2
c(J..(A
~
+ ( - 1)
k
n
X - k- 2
k'
•
(k+l)
S..LA
-~
+ ( - 1)
2k
2k A.n - - 2
(2k+l)
S..LA
(2k)J-~
+....
It will be easily verified that this development is a special case of developments (36) and (37) for 1J..(~
f
0 and (Li,i+l (i
= 1,
k - 1; s
= 1)
or
(s :: k ... 1).
Let us now note that in the relations (36) and (37) the last member that contains the lowest power of A will be of the same degree in A for all ~~(A) belonging
to a s arne field L. . (s
1,1+S
<>
0), bee ause this degree has for an exponent
n - s - tk - 1 (s ~ 0), where t is the largest whole number such that n - s - tk
- 1 ~ 0, and this exponent depends only on n and s.
This shows us that one c an write
¢
s-l rA
k
..(~ (A) .. A
'"t"..(~ (A)
in case n is divisible by k.
division of n by k is v, 1
~
.
1f cp..(~
(
Li , i-s, s
» 0,
When n is not divisible by k and the rest of the
v
~
k - 1,
-3)-
q> ..<~(Ak)
for v
>
Ak+v - s- l <p~(Ak)
for v
*s
v S l
A- -
s
,c2Q
""!"'
C Li i +s
s ~ 0;
"
(41)
ep ..<.~(Ak)
v s l
A+ -
v s k l
k
A + - - <:P..<.j3(A )
~
k
In all the cases 1f"..,..) A),
for v +
B -
l<k
,
for v + s - 1
~k
(()~A
C L.J., J.-8,
.
'i'~
S
>0 •
k
k
C{)43(A) designatepoJ:;yncmialswhose argument is A •
In particular we have this consequence of Theorem XXIX.
XXX.
If tre. ~ .!!. of matrix ¢, considered in XXIX, is not divisible
and the rest of
~
division of n
~
k is v,
k+v-s-l
= AQh
when
f""'B C. Li,i+S
~
would
!?l
k
~
¢J{3(1) for v~ s
and S ~ 0, and
rJ.
P~(AOh
)
= Av+s-l
Oh
k,
p"<{3(l) for v + s - 1 <"k,
rJ.
v+s-k-l rJ. (
:: ~h
p..(~ 1) for v + s - 1 ~ k
When{J..(p <. Li,i_S and s
> 0;
k
AOh is one of the roots of the equation A - 1 = O.
If n is divisible by k, one will have
¢~../>Uh) = ~~l ¢..(..(1);
¢""'{3(AOh )
= ~~S-l ¢~(l)
for
¢.,z;3(A0l1 )
= ~~l
fortp..({3 C Li,i_S, S ;7'0.
¢~(l)
qo~
C
Li~i+S,
s
~ 0;
-'1J.Note.
The rules that define the factor in front ofP.,(~(l) in the right parts
of the equality of this theorem are a little complicated.
But they can be re-
placed by a single equivalent rule that is more convenient in the applications
of theorem XXX and which develops immediately from it.
¢.,(,,( O'Oh)
=
~~l
= ~~l
¢.,(,,((1) for n
=- v
It is the following:
(mod k),
¢.,(,,(l) for n ; 0 (mod k);
for ¢.,(~ (Xah) these exponents v - 1 or k - 1 remain
invariable as long as ¢..<p (A)
is in the same field as ¢..t. JA) and increase or decrease in s units when ¢..<l3 (A) is
displaced from s fields to the right or left ¢~(A), provided v .: s - 1 or
k + s - 1 is replaced by the ne arest positive number of one or the other of these
numbers and congruent to them modulus k)if one or the other of them become s larger
than k or negative.
Then, if one imagines matrix ¢ written in the form:
the exponents of A of the equalities in XXX are respectively
Oh
k - 1
0
1
•••
k - 2
k - 2
k - 1
0
•••
k - 3
•••
k - 1
... ... .. ..... . ..
o
1
2
-35-
=
when n
0 (mod k) and the values
v - 1
v
v + 1
•••
v+k-1.
v - 2
v - 1
v
•••
v + k - 2
v - k
v - k + 1
v - k + 2
.... ... .....
........ ......
v - I
•••
when n '; v (mod k) on condition of reducing the negative or positive numbers the
ones larger than k of this table to the smallest positive numbers and which are congruent to their modulus k.
XXXI.
Let A
= Al f
1 ~ ! ~ of ~ stochastic matrix
composable, nonprimitive and
~
the index k.
Then for!!:!. .t..
¢
that is not de-
= l,n
(42)
.."
rlhere
2
signifies that
~ ~
is taken
f2!.
all the values of
~
corresponding
~~~
to any: field L of
ih
¢.
In fact, by supposing that
¢ is of the form
(22)~one can write the system
Al X = X¢ l.Ulder the form
This system admits a non-null solution
Xl
so that
= (Xl(1) '
(1)
••• , x t
1
), ••• ,
Xk
=
-3~
t
1_X~(2)
"J:
1
~
,c
x~l)
__
&:J
-<=1
¢-<, <11
~ +
t:l
t'
and so forth.
One has obviously
t
3
~~l ~tl+.(,tl+t2+~
(-<
• 1
= 1,
2, ••• , t~)
etc, then
from which
2
~
Ai,LJX
(3)
.~ (1)
r=~x
,
3
~
"'l..GJ X
(4)
"\; (1)
=&;tX
,
..., "'k-1
1
.--1
..,Gx
(k)
~,
=.4x
(1)
and finally
Aik
But Al
~
f.
-::: (k)
~x
~
1, consequently ~x
(k-l)
0
./". X " " .
(k)
.,.
. ~ (k)
~X
•
= 0, then ~x
71
(1)
-:-l
= O,..zx
(2)
... 0, ... ,
-31(1) '
(1) '
..• ,
(k)
xt
are
Let ~ ~ ~ matrix considered in XXXI.
Then
By noting that the numbers xl
.
proport~onal
to the
k
~~l(Al)' ~~2(Al)' .", ¢~(Al) one gets from the preceding equalities relations(42).
XXXII.
X
z
(43)
(i = 1,k),
In fact, the system X = X¢ has in this case, at a factor that is almost constant, a unique positive solution
where
(h»)
• • • J
xt 0
(h =
I;k)
h
are the solutions of the system
As in the preceding theorem it is concluded from this that
from which
(~ =
I";ii).
Now let us consider the system Y = ¢ Y that admits, at an almost constant
fa.ct.or, a unique solution
= yno =
1,
-38from which
(45)
(p = l,n).
From these relations and from
(44)
it is seen that
(46)
(i
=
l,k)
(i
=
l,k),
then
A being a constant number.
However
n
'C1
2
J..=1
·
¢.I..,( (1) = ¢'(1),
from which
¢'(1) = kA, A
which proves
=
¢~(l)
,
(43).
For the matrix
¢ that we are considering here, by depending upon theorem XXVI,
we can write
where
I~I <.1, ••• ,
'alJ.l-<.l and, k + IJ.k + v = n.
It is seen from this
from which
(i
=
1';k).
-39One sees further that
because ¢.,(p«l) are all positive.
XXXIII.
has
Aa
Let ~ suppose ~ the matrix ¢, stochastic and of the order n,
= 1 for zero of
~l1tiplicity
o
.... .
¢ =
o
o
~+1,2
~+l,l
...
k and that it is placed in the form
• ••
o
o
o
·..
o
o
o
·..
• ••
• •
• ••
where all L.,(p«co(
l~
= I;'V)
...
co(
o
o
..
o
·..
o
·..
Lk+l,k ~+l,k+l 0
... . ...
.. .....
Lvv
Lv,k+l ~,k+2 ·..
~ ~decornposable
¢~(A) ~ 0 for every
.. .
·..
·..
o
o
and ~l' ... , L ~ isolated.
kk
that belongs to fields L (9
99
Then
= I;k) and for every p such
that ¢~ does not belong to one of the fields L99 ;
o
One deIJ'!.0nstrates I. by noting that under the conditions imposed on co( and
p,
determinant ¢~(A) will have one band of elements that consistsof t g lines and
will contain only t g - 1 non empty columns.
This band corresponds to field L
99
in which falls or in other words, to which co( belongs.
By developing ¢..<p by means
of the LaPlace theorem following the minors of the order t g (order of L ) formed
99
of aLements of the band considered, one sees that all of these minors are zeros,
hence ¢..<p (L) ;; 0 for co( and ~ set forth in the statement of the theorem.
the
Next, No. 2°i8 true when ~ and ~ satisfy the conditions of No. lObecause then
¢~(A) s
O.
In the cases where these conditions for ~ and ~ are not satisfied, we
have two possibilities: either
LQQ(Q .. I,;1{);
~
and
~
belong to a same one of the fields,
or ~ and ~ do not satisfy the conditons of No. 1° and do not belong
to a same one of these fields.
Let us consider the first possibility • Suppose
LQQ , I ~ Q ~ k.
~
and
~
belong to field
Then by designating by L~ the matrix that is obtained by removing
in LQQ the oolumn
~
and line
~,
one will have
from which it is concluded immediately that
Let us pass to the second possibility.
to one of the fields LQ~' 1 ~
h
= 1,
2, ••• , v - k;
Q ~k,
2)..( and
~
Two cases are presented:
1) ~ belongs
and ..( belongs to one of the fields Lk+h,k+h,
belong to a same one of these latter fields or
to different fields ~+h,k+h and ~+g,k+g'
In the first case we will have in .0-<p (A) one band that consists of t Q
lines and of t Q columns non-empty,
to which ~ belongs.
Q
being the index of the field LQQ , I
~
-
I
Q ~ k,
By using LaPlace's theorem and by developing ~~(A) according
to the minors of this band wc;·w!ll sot ¢..l,.f3(A).in tho form.
from which one sees that No. 2° is true.
In the second case one easily sees that
-41A()..) designating a certain factor that does not vanish generally for).. = 1.
0
concludes again that No. 2 is true.
also has ¢(k-l)(l)
..q3
Furthermore, one sees that in this case one
= O•
Thus our theorem is found to be completely
Chapter II.
7.
chains.
One
Generalities.
proven~
DISCRETE I-fARKOFF CHAINS
By nO'loT we are well acquainted with what is me ant by Harkoff
Nevertheless, for clarity, we recall here to the reader the fundanental
definitions.
Let us consider n mutually independent event s
••• , A
n
and an indefinite sequence of trials in each of which one of these events must
necessarily take place.
Let
be the probabilitic s of A in the initial trial, numbered 0; we have also
2 POk = 1.
The probabilities of A in the trials following the initial one are determined
by the following rule:
in the trial k + l(k = 0, 1, 2, ••• ) A~, 1~{3~ n j has the
probabilitycf43 if we know that A.,I..' 1
~.,I.. ~ n.
has been observed in trial k.
This rule takes us at once to the fundamental relation
(1)
which holds for k
A~
= 0,
1, 2, ••• and in which Pkl.,l.. designates the probability of
in the trial number k when the results of trials numbered 0, 1, ••• , k - 1 are
indetermined.
From
f
t~e definition of ~ we see that
J.¢J.
(2)
~=i -43
and that
¢JJ..f3 ~ 0,
(..<
=1
("'-,p
¢=
:0
1,11).
We again now suppose that in the matrix
rPl l
#12
{J21
t4.2
.• • .
•
if)nl
l,n),
0
• •
~2
¢>In
•• e
...
..• • ..
....
~n
c(Jnn
no column or row is empty, or in other words, each column and each row has at least
one element which is not zero.
This supposition is equivalent to the very natural
condition that in each of the trials, boginning with trial number 1, any of the
events A , A , ••• , An is not impossible, then is not suppressed as soon as we
2
1
depart from the initial trial, and as soon as we do not have among the A an event
which terminates the trials in such a way that as soon as it appears we can no
longer continue the trials.
If all of these conditions concerning
ifJ.l$
are admitted, we see that
¢ is a
stochastic matrix of order n.
We call the matrix
¢ the law of the Markoff chain under consideration and it
will be called simple and discrete under our conditions.
We call the probabilities
4J~ the probabilities of transition of the system of events
A.I.. or the Harkeff chains
under consideration.
We c an also consider Harkoff chains which are simple and continuous, complex
and discrete, or complex and continuous.
But we will put the se aside and consider
here only the simple and discrete chains which are, in every case, fundamental and.
serve as a basis for more profound research and for the applications of chain
probabilities in mathematical physics.
-4}General .8.:llution ~ system (1).
8.
According to (1).
whence, in using this formula for Pk-l} "" and on replacing Pk-l/"" for
f
Pk-2 J 't
%J.. '
we obtain
setting
By continuing this process, we arrive at the relation
(5)
where
We can also verify without difficulty that we have
n
(6 bis)
rf) (k) =
., -«3
.:>' t$ (k-l) CP
i~l
'{ -<t
~~.
We have recourse now to the following formula of M.
o.
and
------------------- --- - ----.- -- -1
o.
1
Perron.
Perroi1, Eathcmatische 1-umalen, 64 (1907), p. 257.
Let
-44be the characteristic equation of A having the roots PI' P2' ••• J P
plicities ml'
~,
• • • J
~
2
.. -
t=l
of multi-
Then, by setting
ms '
n
a(V)
s
n
(v-I)
a"(lSa~~
...
:s
.
~=l
(v-I)
a -<?J
a..(~,
-
( (1) =
a-<f3
~)
and by designating the minor of the determinant of A corresponding to the element
a~..( by g-<{3 (p), we shall have
n
a(v)
(7)
~A
-'I"'
Ir).-l
where D
p
=]
)..=1
1
em..A.-1)!
designates the
D~-l
P
~
- 1 order derivative taken with respect to P and where
(8)
This formula is also true for v
for
.t..
+~.
= 0,
if we place
a~~)=
° for A r~,
and
::0
1
°
In the case where A(p) = has simple roots PI' P2' ••• , P the general formula
n
is replaced by the formula;
n
vg (
a(v) = ~ P"A 13 P"A) J
~ "A=l A'(p)
)..
Let us now take the matrix
zeros
of multiplicities
¢ of the system (1) and let us suppose that it has
respectively, A. , A. , ••• , As being different from
2
1
us
s
~(k)=
(10)
.,(~
'2
i=O
m.-l
D 1.
A.
1
(mi - I)!
Ac ;
Then formula (7) gives
1.
[Ak¢"'!l(AJ]
Yi 0..)
A=t...1.
Hhere .¢..<j3(A) designates the minor of the determinant
¢(A.)
= }A.E - .01
corresponding to the element ~.,( and
. .TJri ('\)
1
1\
=
¢(t..)
(A.-A.)
1.
mi
).~)rru('\I\,-'-:l
>.....)~ •••
= ('\A-'I]
By using this expression for
('_'.
1\ 1\1.-1
)mi _l ('\_'. )mi +l
(,\, )ms
,.., /\'1.+1
••• /\'-/\'s
•
rfJ);) we develop from (5)
the fundamental formula
for Pk/ ~:
s
=
(11)
~
&;.J
i=-O
where
(12)
Thus we have found the most general solution of the equations (1).
Further on
we will show another process for solving them.
9.
IvIatrices.
Discrete Harkoff gtlains for the
of Indecomposable and Primitive
We will now examine the conclusions that
formula (11) when the matrix
zeros.
~
loJe
can derive from the general
¢ satisfies special conditions with respect to its
Let us suppose in the first place that ¢ is a nondecomposable and primitive
Then, as we know, all of the minors ¢~~(A) are positive for A~ 1, there-
matrix.
for
Aa
~
/
= 1 will be a simple zero" for ¢(l) = ~ ¢
~
will not have zeros of absolute value 1.
(1) is positive.
Besides ¢
.t,I..
Summing up, ¢(A.) is, in our case, of the
form
and the formula (ll) may be written
(13)
Pk
s
11
I~
=10
1
C-:-1
~
(1)
+.2
~
.
.
i=l (m.-l)'
m.-l
D~
A.
where
(14)
¢=<f3(1)
¢' (1)
,
because
Let us consider~o~(l). ¢~~(l) and ¢I(l) being positive, it is a positive number.
Then the s,ystem Y = ¢ Y certainly admits a positive solution Y~ = 1 (~ = I;n)
which, except for a constant factor, is a unique solution of the system•. Then, in
virtu e of the relations
we have
consequently
whence
]
~=l
¢13(1)
= 1.
¢'(1)
TrIe then obtain
then, setting
we find
,
(15)
pp, p
= 1,n,
being positive numbers such that
n
(16)
~ p~ = 1.
p=l
Finally
(17)
-48whence it is evident that
(18)
lim Pk/ A = PA'
k...:,.oo
t'
t'
sincel ~I <. 1, i
=
r;s,
and ~~ (A), as well as their derivatives, are finite for
A = Ai.
We see that the positive numbers
result toward which the probabilities
p~
are the limiting probabilities or the
Pkl~
tend with increasing number of trials.
A very remarkable fact is that the probabilities
do not depend on the initial
The equalities (15) show us that they depend uniquely on the
probabilities PO..<.
law of the chain
p~
¢.
We can sum up the results obtained in the following theorem.
!!. ~
Theorem A.
consideration, is
stochastic matrix
~~ ~ecomposable
¢,
~ ~ ~ ~ Harkoff chain under
and primitive
~
has zeros
whose multiplicities are
AI' A2 , ••• , As having of necessity absolute value <.1, ~ probabilities Pkl~ ~
dofined by the equalities (17), where Pp ~ given by (15) and represent ~ final
probabilities toward
~ Pkl
P tend
the probabilities Pp do not depend
~
~
the initial probabilities of the events \ .
From formula (17) we easily derive
n
(19)
s
'~'7"l
.-?;
~
6=1 i=l
1
(m. -1) ,
1.
•
the number of trials increases indefinitely;
As a matter of fact, from (6b)
n
n
.7 to(k)
I ",0
==0 If)
~ i
_."--1
~=l'"
((=1
(k-l)
.t.. 't
'
therefore
n
2 cp (k)
~=1
n
71
=
~~ qJ-<{3
=
2p()..( ~ ep(k)
..(j3
(k
= 1
==
0, 1, 2, ••• )
and
n
~ Pkl~
~==1
"n
P ""
""
= 1-
{3
From this relation and (1'») we get (19).
But we can proceed directly and obtain some more det,ailed result,s.
Let Ai be one of the zeros AI' A2 , ••• , As different from 1.
A.X == X~ admits a non-null solution'
1.
and i f ¢..<f3 (Ai) are not all zero, we c an write
(i).• x (i).•••• •• x (i) --
Xl
n
2
~
...
(A.). ~ (,).
P""l .~ • P.t..2 ~i •
But
'" x{3(i)
Ai .2;
~
whence
~ (i)
~x
.t..
e-
.t..
... ~
(i)
x.t..
.t..
<::"I
.z~ CP..<f3
~
(i)
= ~X.t..
,
== 0, therefore
n
'C1
(20)
~ ¢..q3(A.) == 0
p=l
1.
("" = r;=n).
The system
when all of the ¢~(Ai) = 0 this relation is trivial.
Let now AI , A",
IA" I<I,
1A'I <. 1,
be two different zeros of ¢.
The relation
(20) now leads us to this one
= 0,
A' - A"
whence letting A" tend toward A', we see that
if AI, jA'1 <-I, is a double zero of ¢.
By continuing this well known reasoning,
we see that
o
(21)
if Ai'
I ~I
< 1, is a zero of ¢ of multiplicity mi.
In consideration of these equalities, we verify without difficulty tilat
n
;~[D~ ~i~(A~A=Ai
(22)
=0
whence we obtain (19) immediately.
Example 1.
Let us consider a chain whose law is
0.2
0.3
¢ =
0.2
0.7
The characteristic equation of
A-O.)
¢(A) =
- 0.2
- 0.7
0.2
¢ is
- 0.2
-0,,5
0.5
-0.3
- 0.2
A. -0.1
A -
=
(A-l)(A.-0.3)(A.+0.4)
= 0,
-,:I:"
vThence
The general expressions for ¢~(A) are given by the table~
1
0.2A
A - 0.6A - 0.01
2
0.2A
3
0.5A - 0.19
+
+
0.19
0.7).. - 0.31
2
0.08
0.2A
A - 0.4A - 0.32
O.)A
+
0.01
+
A2 - 0.8A
0.08
+
0.11
from which we calculate
1
1
2
0.28
0.28
0.28
2
3
0.31
0.)1
0.31
3
0.14
0.10
¢l (-0.4)
¢2 (-0...4)
1
0.39
0.11
-0.59
2
0.00
0.00
0.00
3
-0.39
-0.11
0.59
The generalformula
(17)
in the case of simple zeros of
¢,
-0.04
as the present case,
can be written
(23)
(k = l,n).
By noting that, in the example under consideration,
¢'(0.3)
= -0.49;
¢'(-0.4)
= 0.98,
we obtain from this formula and from the preceding tables the follovdng expressions
for the probabilities Pkl~:
The final probabilities are
Example 2.
Let there be
¢
=
0.5
0.2
0.3
0.2
0.3
0.5
0.4
0.2
0.4
then
~
= 1;
A.1
= A.2 = 0.1;
~
¢2 (A.)
1
0.2A. + 0.12
2
0.2A. - 0.02
3
0.3A. + 0.01
¢3 (A.)
0.4A. - 0.08
A.2 - 0.9A. + 0.08
0.2A. - 0.02
A.2 - 0.81..
0.51.. - 0.19
~
¢1 (1)
¢2 (1)
¢3 (1)
1
2
0032
0.18
0.31
0032
0.18
0.31
0032
0.18
0031
3
+
0.11
-.53~
1
2
0.00
0.00
0.00
2
0.20
-0.70
0.20
3
0.04
-0.14
0.04
3
0.)0
0.50
-0. tI:J
Pkll =
32
(O.l)k [
nr
+ rr:r(36k + 49)POI -
]
(126k + 32)P02 + (36k - 32 )P03 '
+
63P02
J.
The final probabilities are
10.
The Case of ~ N;)ndecomposable and Itnprimitive 1-1atrix
suppose that the matrix
¢ be
¢. Let us now
.indecomposable and irnprimitive.
Being indecomposable, it admits, as in the case
he
= 1 as a simple zero, and
it is imprimitive it can be, from XXII, reduced to the form
o
(24)
..•
¢ =
0
¢ will be
o
• ••
o
...... •
•
0
0
0
0
supposing that it is cyclic of index m.
equation of
• ••
L
• ••
m-l,m
0
Then by XXVI, the characteristic
-54where v
= 0,
i f n ~ 0 (mod m) and ~ 0, if n -; O(mod m) and \ Cil
But among the numbers C there can be
i
~
<
1 (i
= I,ll).
equal to ai' m equal to a , ••• , ms
2
2
equal to as' in such a way that the definitive form of ¢(A) will be
(25)
That being so, we can, from (10), write
(26)
denoting by A.ch(h
= 0,~1)
m
A-I
h =
the roots of the equation
= 0,
r;mg ) the
m
roots of the equations
X - ag
= 0,
(A-A
)mg
(g =
I;S")
and setting
(27)
CP(}.,)
(28)
= ¢~~)
gh
m
.. (Am _ l)(X .;,
¢ being indecomposable and all
¢l~(l)
,
= ¢2~(1) = ••• = ¢n~(l),
~
= ¢~@ (1) =
~ ¢~~ (1)
~
~)~
••• (Am _ a )
m
S
•
S
¢.,(~(l) being positive, we .find, as above, that
= l,n, whence
= p~
~
0
and
. ' 71
~p~
= 1
~
(haD
= 1).
The number v is fixed and finite and k
may be arbitrarily large.
It follows
from this that
1
(30)
v-I
(v - 1)1
D)"
for k<. v this quantity is not in general equal to zero.
In repeating the reasoning
that led us to the equalities (21), we shall verify that
.-::-1
.z¢~(0) = 0, ••• ,
(1)
(..<. =
I;n),
~
whence
(..<.
for the case k"v;
= I;Ii")
for k~v this corresponds to (30).
We can alro see th'at
(..<.
(3)
From these equalities, we conclude that
m
g
S
;:--,
(4)
L:1
[3
-·71
~
g=l
1
(mg-1) 1
]
h=l
m -1
Dg
)"
(..<.
= r;i1).
= r;ri).
-Sfr
Now with aid of
Pkl~
~PO..(<P~
-C"1
=
(
)
, we can write
OS)
by putting
(6)
s
'V
PI!
1
~ ( 1)'
kl ~ = gel
mg- •
1
~
h=l
m -1
,
~po.l~(A.)]
Dg
A.
..(
),,=A
gh
= (v-I)'
(8)
... ,
mg
.'V
It is evident that Q
kl
v-I
~
.
varies irt'Bgularly when k takes the values 0, 1, 2,
and is alw~s zero for
k = v,
we always have
n
-'u
2J
In the same
kl to'A = o.
Q
8=1
w~,
it is clear that
lim
k"too
P~IA
to'
= 0,
v +
1, •••
Moreover, because of
(32),
n
c...,
24 P~IA
(41)
~=l
which follows from
'" 0,
to'
(34).
There nOvI remain the quantities Pk\f3.
They vary with k in a very remarkable
way.
1et t l , $2' ••• , t m be the orders of the square sub-matrices 1 , 1 , ••• ,
11
22
1
mm
('!Jlich are empty) corresponding to the form (24) for ¢, in such a way that
principal fields ~2' L , ••• , L
have t rows and t columns, t rows and t
ml
2
2
23
1
3
columns, ••• , t
m
rows and t
columns, respectively.
l
Let us now consider the sum
~ PO)-<j3 (A.ah).
When.( runs over the values
1, 2, ~ •• , t 1 , the quantities ¢-<j3(A.ah) remain in the same field 11,t~t being one
of the numbers 1, 2, ... , m and depending on the value of the index
as .( runs over the values
t
t
1
+ 1, t
l
+ 2, ••• , t
l
+ t ;
2
l + t 2 + 1, t l + t 2 + 2, ••• , t l + t 2 +
.... . . . . .. . . . .. . . . . .
+ t
L
•
3,t'
... ,•
m- 1
+ 2, ••• , n
Lm, t f
respectively.
From XXX,
=
m+v-t
AO
tit.
P-<j3
~3;
for v - t ~ 0
f3.
Furthermore,
-58or
where Q = v - t or m + v - t following the cases indicated when
the values 1, 2, ""
ql'
~
takes one of
For the values
¢~~(Aah) will belong to the field
L2,t; then, from the remark made about XXX,
will increase by unity and we shall have
We can continue this reasoning and arrive in this way at the following
result.
Suppose
+ •••
q2
(42)
=
PO ,t +1 + Po ,vl~
0
2 + •••
l
..... . . . . . . . . ... . . .. . .. . . ..
qm
= Po , t 1+ ••• +tn- 1+1 + PO,tl+ ••• +t n- 1+2 +
...
+ Po n.
Then
Noting that in our case
=
.2 ~v-l
¢
Oh
~
~
(1)
= A~-l ¢ 1(1)
"Uh
Q
we shall have, with the aid of (43)
m-1
k
':"I
P'kl ~
=
t"'
~
h=l
':I
A.Qh
~ Po.<
.I...
¢oo<@ (Aah)
¢' (A.Oh )
m-1
~
k..v+1+Q
h=l A.Qh
We have the relations
m-1
'-:I
.24
L
~+~
.
h=O "'Dh
~
(1~ ~ ~ m .. 1;
0,
(~ =
for the mth root of unity.
From the above (we get)
m-1
~ ~+~1
0, 1, 2, ••• )
m-1
= -1,
h=l
~~=m-l
h=l
and
m.. 1
~
k-v+1+Q [
£j A.Qh
ql + A.Oh q2 + •••
h=l
i f k .,. v + 1 + Q + 0 - 1 :: O(mod m).
:s
-:-,
Since ..6 q
= 1,
we have
m-1
(44)
>--~-v+ 1+Q
h=l "Uh
and taking account of (29),
(45)
= mq 'If
.. 1
~ = 0, 1, 2, ••• )
But k - v + 1 +
Q
+ 0 -1 = k - t + ~ or k - t + m +
X, according to the values
of 9 and this number must be divisable by m and (f must be one of the numbers
1, 2, ••• , m.. We aco' from· this that'?> is that one of the numbers 1, 2, .. 0' m
which satisfie s the condition.
(46)
CS' == t - k (mod m),
where t is, in its turn, defined by the relation
(47)
These two conditions completely determine the value
Let us now take the sum
(45)
of Pkl~.
We have
p~ + Pkl~.
Let us examine sorne of the properties of this quantity.
We note first that q~ does not vary when ~ takes the values for which <P1~
does not pass outside the field LIt.
When
~
runs over the values 1, 2, ••• ,n, t,
and consequently 2f run in a certain order over the values 1, 3, ••• , m.
These remarks lead us to the relation
n
m
'-'
~ (p~
:-"1
+
¢(3@(1)
'C"l
Pkl~) = m J:j q~);
t=l
6<:Ltt
¢I (1)
m
= .~ q~ = 1,
r=l
m
-,
2i
where we use XXXII and the relation
~
=1
q
= 1,
as is evident.
Then
-61n
n
~~l(P~
+
prkl~) = ~~ mp~q?J
= 1,
whence we can also infer the relations
Consequently the numbers
mp~q~l'
being positive can be interpreted as probabilities.
And we c an in f act do this.
From XXXII, for our case
= ~'(l)
m
'
that is to say
(49)
= m ¢@@ (1)
=
¢' (1)
•
However, this quantj.ty can be explained as follows.
Let us divide the events A , A2,
l
(50)
Bl
= (Al ,
B
= (A.e
2
.. .
0
1
•
... , An into the subsets
A ,
2
A
+1'
0
•
0'
A.e
ou,A.e
t +2,
l
0
•
....• • .• .
0
0
0
),
1
.e
1+ 2
),
•
which we call (ensembles en chaine) chain sets for as we see from
(24), the
appearance of one of the events of the set B makes possible only the appearance
l
of one from the set B2 in the following trial, which in its turn can only be
followed by one from the set B , and so on, the sets B , B ,
l
2
3
each in one of its terms one after the othero
.~.,
B , is realized
m
We see that in our case the chance
does not depend on the sets B , B , ••• , B , their succession being completely
l
m
2
defined by the first event, but only on the events in each of these sets.
This being established, if we know that after a certain number of trials we
are Je d into the set B corresponding to the field L •
t
tt
as the probability of the event
A~
We must consider
which is a part of the set B that we have
V
designated by the symbol P~'d'tt 0
On the other hand
from the theorem of addition of probabilities, is nothing else but the probability
of observing one of the events B2S' in the initial trial, the set
by
(46)
and
B~
being defined
(47).
We now see that
(52)
q25 PA<.L
t'
tt
is the probability of an event '..j'hich consist.s of the simultaneous realization of
two other events:
1) the appearance of one of the events of the set B(S'in the
initial trial, 2) the appearance of
~, belonging to Bt in the kth trial.
Let us further note that
I
P~ ~ -7 0
and Q ~
kl
==
0 when k..,.
00
through any value s
whatever and that q'6 PA.c:.. L
keeps the same value for fixed ~ and k tending to
j-'
tt
infinity through values of the form k == l:JJll + t + 2f, t being a f~ed number defined
by the condition cp~C ~t and<s' a constant (number) taken from the sequence 1, 2,
000,
m.
Therefore, q<s
p~
t '.
is the final probability of the event
A~
when we
U
know nothing of the results of the trials considered and we consider the indefinite
number of trials of the numbers
t + '6, m + t + ~, 2m + t +0, ""
This probability is changed with changes of ~ and 0'.
k runs over all the positive whole number values,
I'llhen ~ rem~ins constant and
'6 runs pariodically over the values
1, 2, ••• , m, then the probability periodically takes the values
k\~
with additive terms P
and Qkl ~ of which the first tends to zero for kT 00 and
the second becomes zero as soon as k goes beyond a certain value.:
We see that the const ant oscillations of P \ ~' represented tiy the sequence
k
(.53) are periodic with period m.
Finally relying on theorem XXXI, we find the relations
whence
(.54)
-64for by (51)
The relations (54) are also evidently of the same form of
¢ in our case and
from the remark which has been made on the sets B , B , ••• , B •
l
2
m
\fe sum up the most important established f acts in the following theorem.
Let ~ be give~! Markoff chain of which ~ law ¢, ~. inde-
Theorem B.
---
composable and imprimitive matrix, be reduced to the form
¢
0
L
12
0
0
.
..
0
0
.0.
0
L
23
• ••
0
.. • •
0
0
• ••
0
0
• ••
• • •
0
loThere ~ ~ the cyclic index ~ ¢; let
(V>O,
Ia.) <
be the characteristic equation of
1, ... , Jas}<l)
¢. Then the probabilitl
l\:W
~ A~ is
where p~\~ and Qk!P have the values (37) and (38) and tend toward zero with k....,..co,
is the probability of the set
in the initial trial and
is
~
probability of the event
enter this set,
~
A~
in the set B , when
t
~
know that
~
mustE!:.-
numbers ($ and t being defined by the conditions
k ;; t - "'(mod m).
When k ....
CD,
the probability
Pk)~ ~
periodically
~
the values
Q2PA< L ' ••• , Q PA<.L
~ tt
m ~ tt
-:J
.6
Pili
A<L
k ~
~
tt
= 0,
(t
= I,m)
in such a way that
-----
Remark - If we suppose that for the matrix f; considered in thi.s paragraph the
characteristic function ¢(A.) does not contain the factor
v
= 0,
we must, from XXX and by sUPposingq:\~( LIt' take
)..V,
in such a way that
whence g = m - t, and
we conclude from this that the number k - v + 1 + I;j +
be replaced now by k - m + 1
m- t +
oft
have the same value as above.
Example 3.
considered above will
1 = k - t + ~ , that is to say will
Therefore, all our deductions will remain the same
with the single modification that for v
all in the expression of
t -
t -1
Pkl~
= 0,
kl ~
the quantity Q
does not enter at
that will now be
We will consider two illustrations of the theor,y explained in
this section.
Given a chain whose law is
¢ =
The matrix
0
0.1
0.5
0.4
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
¢ is evidently non-decomposable,non-primitive and of index 3; it is
given in the cyclic form and 1
12
is composed of the numbers 0.1,
of numbers 1, 1, 1 and L
of one number 1.
3l
It is easily found that
0.5, 0.4;
1
23
-67-
-0.1
¢(A)
=
-0.$
-0.4
A
0
0
0
)..
0
-1
a
0
)"
-1
a
0
0
= A$
2
.. A •
Therefore the zeros of ¢ are
2ni
),,00 = 1,
hal
=
eT
4ni
,
A
02
e T
,
which are simple, and
A
of multiplicity v
=
=0
2.
Finally we find
~
¢l (),,)
¢2~ (A)
),,2
1
A
2
0.lA3
A4 - 0.9)"
3
0.5A)
4
0.4>!
),,2
5
¢3 (A)
A2
¢4 (A)
A.2
¢$ (A)
),,3
O.lA
0.1A.
0.lA2
0.5A.
A.4 - 0.5A
0.5A.
0.5),,2
0.4)"
0.4A.
).3
),,3
A4 - 0.6)"
A.3
0.4),,2
),,4
-68-
(1)
¢5 (1)
~
¢l (1)
¢2 (1)
¢3 (1)
1
1
1
1
1
1
2
0.1
0.1
0.1
0.1
0.1
3
0.5
0.5
0.5
0.5
0.5
4
0.4
0.4
0.4
0.4
0.4
5
1
1
1
1
1
¢2 (A01 )
¢1 (A01)
1
2
A01
2
0.1
3
0.5
4
0.4
5
Aal
Aal
2
0.lA
01
2
0.5A
01
2
0.4AQI
1
¢3 (\)1)
"'01
2
O.lAOI
2
005A
01
2
0.4AQI
1
¢4
¢4B(A01 )
"01
2
0·1)"01
2
0.5AOl
2
0.4AQl
1
¢5 (A01 )
1
001A
01
0.5A01
0.4A
01
~1
¢~~(Aa2) are obtained from¢~(Aal) by replacing "01 by "02 and ~~~(O) are all null.
By means of these results and from the general formulas we find that Qk\~= 0
for k
~
1 and
Pk/l
I:
Pk/2 =
10
30 [_skql + sk+1 Q2 + sk-1Q3 J- '
~O
[sk_lQl + skQ2 + sk+1Q3] ,
where
q2 = P02 + P03 + P04'
ql = POI'
sk
k
k
k
AOO + "01 + "02·
=
The sums sk are null for k = 3j.J. + 1
for k
= 3l.J., l.J. = 0,
table of values of
1, 2, •••
Pkl~
crr 3j.J. + 2, l.J. = 0, 1, 2, ••• and equal to 3
The table of
311- + 1
Pk /1 = ql
1
Pk / 2 = 10 q2
Pkl3
=
Pk /4 =
These values comply well with
3.
gives us therefore the following
fa q2
~O
q2
We have ~
=
4,
ou~
3l.J. + 2
q2
q3
10 ql
1
10 q3
IO q2_
10 q3
1
, ,
4
10 ql
4 .
10 q3
q2
Pkl' = q3
'0:0
Pkl~
for these forms of k:
k = 3l.J.
and find P,I4c
q3 = PO,;
ql·
general results.
1J14(
1
For example, take k
12 , therefore t '" 2, t - k
Therefore
4
'" q •
3
We find the same value from the formula
30
=
=,
-3, from which
-r.oon placiXlg in it k
sk+l ::;
8
=5
and in noting that sk_l ::;
8
4
::; 0,
sk::; s5 ::; 0,
6 ::; 3.
For k ::; 0 our formulas give POI~ ::; Po~(~::;l,s), i f we observeth~ in this ~a.se
kl ~
Q
is not generally null.
In reality, in our case we have
therefore
etc.
Example
¢
4.
Take also the mgtrix
0
0.3
0.7
0
0
0.4
0.6
0
0
0
0
0.1
0
0
0
0.6
0.8
0
0
0
0
0.3
0
0
0
0
::;
-71which is evidently non-decomposable nonprimitive and of index 3,
is divisible by 3, it is not necessary to have the zero )., .. 0,
Since its degree
Actually we. find
from which the zeros of ¢
2ni
).,01 = e
3
,
The ~.(j3().,) are given in Table A,
Table A: ¢..4.j3().,),
P
I
,.,
"",=1
3
.:.
2
1 )., -0,5).,
2 0•.52.5).2
0.65).,3-0,15 . .0, 4).,3+0.1
0.3.5).3+0•17 5 0.6J-O'f>7.5
2
2
4
0.36).
3 0.3).4+0•06)., 0.4>. -0.04>. ).,.5_0.64).,
2
).,5. 0 • 33.5).,2
4 0.7).,4-0.035)" 0.6).,4+0 •06.5)., 0.665).,
3
5 0.45). -0.015 0.4>..3+0•035 0.1).,4+ 0 ,)35)., 0.6).4_ 0 •165).,
6 0.55).,3+0•04 0.6).3. 0 •01 O. 9).,4-0.)1), 0.4).4+ 0 •19 ).
0,5).,2
).,.5. 0 •475 ).2
¢~(1)
are given by the equalities:
¢"",1(1)
=
0.5, ¢"",2(1)
=
0.525,
¢~3(1) = 0.3 6,
¢"",6(1) = 0.59
6
5
4
0.2).,4+0.)).,
0.7)\4·0.2).
0.8).,4_ 0 • 27 .5)., o. yt+o. 225>..
0.38).,3-0.02 0.33).,3+0•03
0.62).3+0•0 4.5 0.67>.3.0•005).,5- 0• 565 ). 2
0.435).2
0•.59).2
).5. 0 •4H.2
¢~(1)" 0.665,
(~
¢~5(1)
= 0.435,
= 1, 2, ••• , 6).
We get from them
= ¢11(1)
P1-,
~ ¢.,(..(1)
_ 500 p = 525 p .. 360 p _ 665 p _ 435 p _ 590
- 3075' 2 3075' 3 3075"' 4 - 3075' .5 - 301)' 6 - 307; •
-12We find ¢..<j3 (\n) from table B where each line has a numerical factor common to all
the members of the line and placed at the side.
Table B:
1
.t. ;::
2
3
4
5
¢.t.p <A"Cn)·
factor
6
1
2
>"01
AQI
1
1
AnI
hal
2
~1
~1
1
1
>"01
\>1
X 0.525
3
Aol
>"01
1
1
X003CO
4
A.a1
>"01
1
1
X 0.665
5
1
1
2
>"01
X 0.435
6
1
1
~1
~1
~1
X 0.590
2
~1 >"01
~1 ~1
2
hal
>"01
A.al
>"01
X 0.500
By replacing the values ¢.t.j3(ha1) Aal with Aa2 we get ¢..<~(Aa2)'
¢~(O") are found
from table C.
Table C: ¢.t.j3(0"),
.t. ;::
1
2
2
3
2
4
1
-0.525 0"
2
0.525 0-
3
-0.05250"
-0.050-
2
-0.6650-
4
0.05250"
0.05
0.6650"
5
0.02625
0.025
0.33250"
-0.180"
6
-0.02625
-0.025
-0.33250"
0.180"
2
-0.5 00.5 0-
2
-0.16625
0.09
0.295 0"
-0.21750"
0.16625
-0.09
-0.295 0"
0.21650-
-0.0295
0.02175
0.0295
-0.02175
2
0.4350"2
2
00360"2
-0.300
2
-0.590"
0.590"2
We get ¢.t.j3 (~1) and ¢'13 ()'12)' by replacing in the table C
0
).11
I:
1,
>"11
= >"010",
222
>"11 = >"010"
and
0
>"12 =- 1,
re spectiwly•
6
5
).12 ;:: Ao 20"'
2
>"12 ;::
2
AD 20"
2
0°
-0.4350"
= 1) 0"
and
2
0
2
by
-73•
Now calculate the
•
quant~t~es p~
+
J'~ k'~
Pkl~
by taking into account that ¢' (Aah) :: 3.075
=
~O AOh ~p()..<
~h.
¢13 (Aah)
¢1(A )
Oh
The values found for ¢..q3(1),
¢..q3 (Aal) and, ¢..q3 (Aa2) give
where
2
" A
P~l
t'
::
~ ,k
1\1h
L.J
h=O
'2 P()..<
kI
.i(
we find:
p"
kll
k
(J
= ""'30=7"""'5-
[Sk(5 25P01 - 500 P02) + sk_2(166.25P03 - 90P04) a-
2
1
-sk_l(29)Po5 - 217·5P 06)cr- ] ,
-74...
k13
P
..
k
--.;;.(J_ _
3075
[Sk_1(5 2 , 5P01 + 50P02)(J-1 + sk(665P03 - 360P04)
+
Sk_2(29.5P05-21.75P06)(J-~I,
k
Pk/5"
3~75
[Sk_2(26.25P01 - 25P02)(J-2.sk_l(332,5P03-180P04)(J-1
I =- Pilik l'
P"
k 2
By adding up P~ + P
having the form
,
~ and P~I ~ we finally get Pkj ~.
kl
+ 1,
3~, 3~
3~
These probabilities for k
+ 2 are:
_ 500
~l.L
P31J.11 - I025' q1 + ~ (525P01 - 500P0 2)'
500
P3l.L+ 1 / 1
=~
P3l.L+ 2 / 1
= I08
500
_ 360
q3 -
~lJ.
I52>
(295P05 - 217 ,5P0 6)'
~l.L
q2 + I02';' (166,25P03 - 9OP04) j
P3~11 - 1025 q2 +
(J3l.L
i:02!
(665p01 - 3 WP04)'
360
(J3l.L
P31J.+113 = 1025 q1 + i08" (5 2 , 5p01 + 50P02)'
I
P3l.L+ 2 }3
~
=~
(J3l.L
q3 + 1025 (29,5p 05 - 21. 75P0 6);
_ 435
(J3l.L
P3 l.L15 - 1025 q3 + mg (590P0 5 - 435P0 6)'
P3 l.L+1 I5
.. 435
~l.L
1025 q2 - 1025 (33 2 .5P03 - 180P0 4)'
_ 435
~l.L
P3l.L+2/5 - ~ q1 + i02> (26, 25p 01 - 25P02)'
0
-75-
II.
law
~
of the Decomposable Matrix
¢.
Given a Markoff chain with the
¢ which is of the form:
(55) ¢ =
L
ll
0
••• 0
0
0
•••0
0
L22
••• 0
0
0
•• 00
.• ..·....• .
Lmm
• • •
0
0
L
m+l, 1
L
1
Lm+l,m+1 0
m+l,2 ••• Lm+,m
• • • • •
LV2
0
•
0
........ •
• o.Lvm
..•
,
••• 0
••• 0
• • • • • •
Lv,m+l Lv,m+2 ••• Lvv
where all the diagonal element s L..(.,l.(;,,( =
I;V)
are non-decompo sable &'1d the fields
L , L , ••• , L
are isolated, the rest· of the diagonals not being so.
22
ll
mm
This
matrix ¢ evidently has A. = 1 for zero of the multiplicity of m.
O
Let ~l' t 2 , ••• , tv be the orders of the square matrices L~(~ =
We divide the events AI' A , ••• , An into groups
2
·............ ..
~
(v) (
)
A = All+ ••• +lV_l+l, ••• ,An.
Then we can make the following
remarks~
I;V).
-76If the initial trial leads us into one of the groups A(l), A(2), ••• , A(m),
we shall stay there for all the following trials. Becuase of this these groups can
be called isolated groups.
A(m+l),
u.,
If the initial trial leads us into one of the groups
A(V), or A(t),
t~m + 1, we shall stay there as long as the following
trials give only the events of this group.
One can, of course, leave this group
in order to arrive at one of the higher groups A(t+l), A(t+2), ••• , A(v), because
the pr'Obabilities~ are all null for
We can only pass from group A(t) to one of the groups A(l), A(2), ••• , A(t-l),
which will happen if in one of the trials that follow that which lead us to A(t),
we will have one of the events of groups A(l), A(2), ••• , A(t-l); this is possible
because according to our supposition, L is not isolated, then L , L , ••• ,
tt
tl t2
Lt,t-l are not all null.
Once one has arrived at a lower group A (tl~ t l " t"
we can only in the following trials sta,y in this group or pass into a group lower
(t )
than A 1 ,this is possible only for tl~ m + 1. And so forth. If a trial leads
us to one of the isolated groups A( 1 ), A( 2 ), "., A(m)" we will st ay ril..ways there
as is noted above.
One sees that the groups A(m+l), A(m+2), ••• , A(v) are transitory
groups and that one always passes them in one direction; groups higher than the
lower groups.
The last remarks are clearly explained by the theorem according as one would
be able to invoke the law of reduction of chains with decomposable matrices or
simply decomposable chains.
Theorem C.
t~m,
The probability of alwa.ys remaining in a transj tory group A(t),
tends toward zero when the number of trials increases indefinitely.
-77Let
,
US
probability of
avent
\
=tl
+ ••• + tt_l+ l , ••• , $1 + ••• + tt) the
th
of group A(t) in the k
trial calculated under the con-
designate by Pk\h(h
dition that in the preceding trials only events of the same group A(t) were
observed.
We will show that
for k -+
(56)
CD,
if A(t) is one of the transitory groups, a thing which is equivalent to the supposi-
tion that the matrices L-I-l'
L-I-2'
••• , Lt, t-l are not all null.
IJ
IJ
We have
2
g
and so forth.
1
(P,
l(l)
ggif g h
I
'
Finally
The elements ~h vThich intervene in these formulas are the elements of the
sub-matrix L •
tt
Let ¢(A) be the characteristic function of L •
tt
Then all the
roots of the equation J(A.) :;: 0 have smaller modulae than 1 as follows from
theorem IX-bis, the sub-matrices Ltl , ••• , Lt,t_l not being all null.
Then one
can write
where o~ r ~ 1,
I Pll<r, ••• , If
~1 < r, m~l, (j~ 0 (one should take into account
that L is not decomposable).
tt
One obtains from it by the Perron formula
m-l
f1)(k) =
I
gh
+
2
k
.,(=0
AO..<
m
Il
~
t =1
2
~=1
1
( llln - 1) I
p
•
by designating with ¢gh (A) the minor of ~(A) corresponding to the element
by y~ iA) the quotient
and by AO..<. and
~¥
m
m
the roots of the equations A: - r
This expression of
~(~£
shows us inunediate1y that
iiJ (k) -?- 0
for
"" gh
from which by
(57) one arrives at (56).
k
-+-
(lD ,
=0
m
m
and A. -f ~ = O.
~g'
Our theorem also remains true if we consider the probability of always remaining in A(t), t> m, after having arrived there in any trial whatsoever that is
not
initial~
In fact, this supposition is only the equivalent of a change of
initial probabilities POg into (57) to the probabilities of A..t+l' A.,(,+2' ... ,
in a certain trial where the passage of a group higher than j\(t) is carried
A-'r+'t
out
~
the group A(t).
The theorem demonstrated leads uS to the conclusion that for k rather large)
one can) with a probability as little different from certitude as one wishes)
affirm that we shall be lead definitely to one of the isolated groups A(1), A(2),
••• , A(t) if chance did not do it at the beginning, or in other words, that the
chain considered will be reduced to another chain that cont ains only the events
of this isolated group to which we will be lead.
In fact we supposed that the
sub-mat rice s
L
m+l,m+l,
••• , L
vv
are not isolated, then for each of them the theorem C is true and as one passes
from one of the groups A(m+l), A(m+2), ••• , A(v) only to a lower group, one sees
easily that our conclusion is right.
Once arrived at one of the A(l), A(2), ••• , A(m) groups, we will
main in this group.
The corresponding matrices
~l'
alw~s
re-
L , ... , L
are supposed
22
mm
isolated and non-decomposable, then we will have from then on one of the cases
studied in the preceding numbers.
Nevertheless, it is interesting to study the
probabilities Pkl~ for the law (55).
We are going to do so for the case where
the sub-matrices L , L , ••• , L
are primitive.
ll
22
mm
Let us first establish a few auxiliary theorems o
··80XXXIV.
Let
+ •••
Theil, for ~ ~ (55), the quantities
... ,
~
the solutions
~
systems
X ::: X ¢ and
that
~
~
has identically
A(h)(l)
(59)
Y::;: ¢ Y,
~
=
From the effect of the theorem of equations with finite differences one
!mows that
k
• 1 ,
(1)
... , A(m-l)
..<{3
• lk
are the solutions of the equations with finite differences
One sees from t,his immediately that the relations (58) are verified identically"
-81The demonstration of the identities (59) is quite parallel.
XXXV.
~ ~
~ ~~.
22
(~l) )
A..«3
(1
(l)(l) A(2)(1)
A;.<~
, ""~
, ... ,
(60)
L
matrix (55) the quantities
, •• " L
mm
~ quantities A""l3 (1) ~!!2i generally
~
and in the fields that
_
ab_o_v_e the fields L 1
-
1
m+ ,m+ ,
••• , L
LU:!ll in the fields LIP
included in the vertical bands situated
and are null in the fields L 1
VV -
-
-
-
-
·1 ••• ,L •
vv
m+ ,m+ ,
Finally, ~ ~ quantities A;.<~(l) have constant values ~ each of ~ lines
of the fields L
l~
ll
, L
22
, ••• , L
mm
•
The quantities (60) represent evidently linear and homogeneous expressions
of the quantities
¢..(~(l), ¢~~(l),
••• , ¢(.;2)(1), that are null for
..(,~ = l,n
(by XXXIII), then the quantities (60) are all null.
2:
Consequently,
¢(m-l) (1)
:<j3
and by recalling the demonstration of
2°
of theorem XXXIII, one sees that the
quantities A",,~ (1), that are equal to
tm
1
-I)!
¢(m-l) (1)
:<{3
,
are not generally null in the fields L , ••• , L
and are null in the other
11
mm
diagonal fields.
()
3.
The system Y
=¢
Y, 1..rhose solutions .are ~(l) <.:an be 'Written as follot'\1s:
-82-
(61)
+ ••• + L
Y
m+l,m+l m+l,
o •• ,
For great€r simplicit.y in t.he reasoning that follows we shall write the matrix rjJ
in the form
o
Lil
o
(55bis)
• •
rjJ =
.• .
•
0
0
0
,
L'm+l,2
L~ll
o
o •
o
•••
0
o
o
o
o
..• ... .• ....
0
0
L'm+l,3
0
Lmm
•• 0
...
L'm+l,m
L'm+l,m+1
by placing in particular
L'm+l,m+l =
0
0
Lm+2 ,m+2
0
0
...
0
......
Lv ,m+2
L'
L' 1 2
m+1,1, m+ J ,
o ••
•••
,
~
•••
Lv ,m+3
have parallel signific ances.
The system (61) is written
(62 )
+
.0.
+ L'
m+l,mYm + L'm+l,m+lY'm+l o
-83One finds one solution of this system by placing
(63)
Y~ = (y Th = a,
(64)
(a
= arbitrary
(65)
° °
0, Yh+1 = , ••• , Yo
m = 0.'
• • • J
... , Y It h
= a)
h
constant) and by solving the system
Y'
m+l
= L'm+l,h
yO + ••• + L'
h
m+l,m+l Y'm+l
In fact, the quantities (63) and (64)
(h is one of the numbers 1, 2, ••• , m).
satisfy, of course, the equations
and the system (65) has a single determined solution because its determinant
IE - L~l,m+ll is different from zero
and positive for the roots of the equation
lEA. - L~+l,m+ll
=
°
are all absolutely less than 1 and the matrix L' 1
1 is non negative.
m+ ,m+
Let us consider on the other hand the quantities A.,(~ (1) for ~ representing
the number of one of the lines of~.
But the system Y =
h
~
indicated above because
They are solutions of equations Y = ¢ Y,
Y does not have any other null solution than the solution
h
1nh is
non-decomposable.
Now, in the line of
the correslJonding qu.arrtites A.,(~ (1) are necessarily equal among -them.
~,
all
-84This consideration is applied, of course, to each of the fields Lll , ••• , Lmm,
then the property of A..(~ (1), shown right away for Lhh is observed for all the
fields Lll , ••• , Lmm"
Let us study more closely the quantities.
(66)
A..(~ (1)
~
Let
1
= em-l)!
be the number of a line
of~.
Then as we have shown
A..(~
(1) keeps
the SaIne value when ..( takes values equal to the numbers of the columns of
Irm
and is null when ..( is the number of column of the fields
"Then .t.. is tho number of a column Lm+
I 1
,m+1, we have
by
p(~)
the constant value of A-«3(l) on the
the value of
A~(l)
Y~+l'
"'I-'
line in
(1)
I
O.
Let us designate
~h (l~h~m)
and by
p~s)
on the same line and column s(s=t + 1, t + 2, ••• , n).
it develops from our reasoning that
whole
~
A..L(.l.
Then
p(~)constitutes the whole Y~ and p(;)the
since these two series represent the only solution of equations (65).
In these equations by considering
Y~l
o
as unknown and Yh as given, one easily ob-
tains the expressions of the form
( 67)
(s= t + 1, t + 2, ••• , n; t = t l + t 2 + ••• + tm)'
where c (h) are definite positive constants (this comes from the f act that all the
s
minors of
lAE - L'm+1 ,m+1)
are positive for ~
= 1.
Now, let Al''''2' ..., AS and ~, m2 , ••• ; ms be zeros of ¢ that are different
from zero ar~ their multiplicities such that
i = 1,s.
Then) by taking into account. theorems Xx.,'CIII and XXXV, one can write
Pk!R c:
~
(68)
(qh
n
+
2 po~ c(h)
~
~=t+l
\
)
p(h)
~
s
1
.~
+ Ld ( m-.,"'-l).-:,g=l
g
•
where
(h
=
I,m),
and
S
71
( 69)
=
2;
gc:l
One sees from this that for
when
~
is an event of group A(h)
k~oo,
(h = 1, m) a.'1d
Pkl13~O
(m+l)
(v)
when A is an event of groups A
, ••• J A , conforming to theorem C.
13
-86The following property of
XXXVI.
p~h)
can be shown easily from:
One has
(70)
(h =
I,m).
n
--:;1
In fact"
..i:J Pkl A = 1, then
~=l
t'
because as is easily demonstrated,
n
s
<~
';::I
2i
~=l
~
gel
The identity (7:1) takes place 'Vrhatever P ,,( m8Y be.
O
Let us then make Po..<
all values of ,,( except the values
then qh
=1
and
(71)
gives us equality
(70)
that we had to demostrate.
-:::-1
Let, us also consider the sum ~ PkfA.
~"'~
t'
n
'~
(h)
Of course,
=q + ~ p c .
h .,(=t+1 Q..(.,(
=0
for
-87One sees from this that
= l~
n
2
Consequently the positive numbers qh +
po..<c ~h6an be interpreted as the final
( )..<=t+l
probabilities of groups A(1), ••• , A m oo1culatErl en the supposition that nothing is
known about the results of trials with unlimited numbers.
As to the numbers
p~h)
they represent the final probabilities of A when one knows that the unlimited
p
number of trials should lead us to group A(h).
As an illustration of the varying results acquired in this number
Example 5.
we shall consider a chain whose law is represented by the matrix
¢
=
0.)
0.7
0
0
0
0
0.6
0.4
0
0
0
0
0
0
008
002
0
0
0
0
0.1
009
0
0
0.4
0
0
0
0.4
0
0
0.3
0.1
0.3
,
One calculates for this matrix
¢(1) = ¢1¢2¢3
~ (12-0.71-0.3)(12..1.71+0.7)(12-0.71+0.06)
= (1-1)2(1+0.3)(1-001)(1-0.6)(\-0.7);
~0(1) = 14
2
3
- 1.lA + 0.13A + 0.0451 - 0.0126,
¢1(0.1)
= 0.0972, ¢I
¢'(0.6)
= -0.0072, ¢'(0.7) = 000054.
(-0.3) = -0.6084,
-soThe minors ¢.,,(~ (A.), their values for the different zeros of ¢ and the values
.
of ¢~(A.) are given in the tables below:
j3
1
.0
¢lp(A.)
(A.-0.4) .0 2.0
3
2
¢3~ ().)
(A.)
.04
(A.)
0.6 .0 2.0
3
0
0
2
0.7 .0~3
(A.-0,3)¢2¢3
0
0
3
0
0
(A.-O o9)¢l¢3
0.1.01.0
3
4
0
0
5
0
0
0
0
6
0
0
0
('I
~
1
0.2¢1¢3
0.4(A.-Oo3) (A.-0.4)¢2
¢2j3 ().)
(A)
0.12(A.-0.4)¢2
O. 28(A.-003 )¢2
0.084.0 2
3
(0.06A.-0.052).0l
(A.-0.4) (0.3A.-0.26).0l
4
(0.02A.-0.004)¢l
(A.-0.4) (0.1A.-0.02)¢1
.02
(A.-o.).0l
6
¢1~ (A)
.0
¢5S (A)
2
5
j3
(A-0.8)¢1¢3
0.2 ¢1 ¢4
¢J,Q (A.)
¢4@ (A.)
0.3 ¢1 .0 2
(A.-0.4) ¢1 ¢2
¢5@().)
¢6j3(A.)
1
0.0648
0.0648
0
0
0.0504
0.0216
2
0.0756
0.0756
0
0
0.0588
0.0252
3
0
0
0.0468
0.0468
0.0104
0.0312
4
0
0
0.0936
0.0936
0.0208
0.0624
-89¢~5(1) = ¢~6(1) = 0;
~
= r;t).
¢3@ (-0.3) ¢413 (-0.3) ¢513(-0.3)
¢C}3(-0.3~
~
¢113 (-0.)
¢213(-0.3)
1
-0.3276
0.2808
0
0
0.2184
-0.1092
2
0.3276
-0.2808
0
0
-0.2184
0.1092
:: 1,;6;
~
= 3, 4, 5, 6.
¢~~(-0.3)
.0 2 (0.7)
¢1 (0.7)
P
~
= 0;
.0 (0.7)
3
.0 (0.7)
4
.0
5
(0.7)
.0 (0.7)
3
0
0
-0.0036
0.0018
0.0030
-0.0045
4
0
0
0.0036
-0.0018
-0.0030
0.0045
~
~
¢~(0.7)
P
1
2
3
4
5
6
= 0;
= 1;6;
p
¢ (0.1)
155 (0.1)
0.1296
-0.3024
0.1656
0.0072
0.3888
-0.)888
.0-«3(0.1) = 0;
~
:: 1;4;
~
0000096
0.00336
0.00576
-0.00288
-0.00432
-0.00288
0.00096
0.00336
0.00576
-0.00288
-0.00432
-0.00288
2
3
4
5
6
= !;be
15 (0.6)
¢5 (0.6)
1
-0.01944
0.04536
-0.02484
-0.00108
-0.05832
0.05832
= 1, 2, 5, 6.
¢~(0.6)
= 0;
.I.. =
r;4, p :: 1;6•
With the help of these table s l"e get the following values of Pkl ~ :
648
Pkl1 = 1W54 q1
S04
+
I454 POS
216
+ ~ P0 6
k
+
(6§al)
+
(O.l)k
372 (1296 P05 - 1944 P06) -
(3276 POI - 2808 P02 - 2184 P05
+
1092 P06)
(0.6)k
72 (96 P05
+
96 P0 6)
= I3
7
23
(
POl + P02 +
+ 27 (0.1
2
:: '3
)k
7
9"
k
POS +
1)
'1.
)
POZ + .L4POS - 7P0 6
'3 P0 6 + (-0.3)(
3'9 -21POl + 18
k
(2POS - 3P0 6) + 8(0.6) (pas + P0 6)'
2
2
(P03+P04+~ros-+3 P0 6) +
(O.n
k
-rcr (12P03
- 6 P04 - lOPoS + lSP06)
One sees that the limiting probabilities are
P2 ::
b-
P 3 ::
'3
1
_ 2 (
P4 -
3"
(Pal +
Po~
~ PaS
+
(P03 + P0 4 +
P03 + P0 4 +
2
9"
2
9"
+
~ P0 6)'
Pas +
'3 P0 6)'
Pas +
2
2
•
:3 P0 6)'
This example showsus that in the case considered in this number the limiting
probabilities depend upon the initial probabilities, as is likewise true for
equalities (68)
0
In terminating this number we still indicate two following properties of
~
XXXVII.
h
Let A.i )
~~
1.
¢~~ (A.~h)
)= 0 everywhere except in the fields
Then
if the zeros of
~
~
sUb-matrix
Lrm
1 L'
1
h :;:; r;iii, and everywhere except in the fields Ll'
,m+, 2 ,m+,.
- --
that
and
~
po
different
~,m+1,
when
L'
when
m+l,m+l,
0 0,
h :;:; m + 1.
If ). (~) is
XXXVIII~
~ ~m+l)
!: ~
of
~
is a zero of L']
-
- -
-
m+ .,lTl+
~
.2 ¢
1 of multiplicity
(m -.t)
~=l -<f3
t
different from 1 and having the multiplicity
-
-------''"-
(
)
(~m+1)
ll~..
_one h_as
IJ
=0
rhe demonstration of thoso properties is immediate and we shall omit it hore.
12.
Another Method for Finding PrJbabilities Pk\~.
ceding numbers are found by means of the Perron formula
The results of the pre-
for~~)
i
We shall indi-
cate now anothor method for obtaining them which is the same as the method that
was used in the study of tho developments of iterated nucloi following the
fundamental functions in the theory of Fredholm I s integral equations.
A.
Let
us n01loT consider first a stochastic matrix
primitive and whose zeros are all simple a
¢ which is
l101i-dBoolrJpo~able,
Let these zeros be
(i =
l,n-l)
-92-
Now let us take the adjoint systems of the linear p.quations
(72)
The second of these systems has an obvious solution
(..< = I;"ii)
and the first a solution
(73bis)
(~ =
l,n; .,( fixe),
which is positive if C;:;' 0 (C i:5 an arbitrary constant).
Since
>u = 1
is a simple
zero of ~ solutions (73) and (73 bis) are unique for constant factors nearly.
result is
(~
Let us set
Then
(74)
Then let us take the sy~tems (g .. 1, n-1):
= l,n).
The
-9.1-
(75)
They are the solutions
(~
= I;i1; .(
(0<
= r;n;
fixet9
(76)
¢ (A)
1..<f3 g
v(g) = C
.(
~ fixe4 ,
where ¢..<f3(Ag ) are not all null because A is a simple zero of
g
¢.
Now let us demonstrate the following theorem:
XXXIX.
For h
rI
g
and
n
-,
o:i ¢.(~(Ag)¢O~(~)
(78)
= 0,
In particular
n.....,
(g :; 1, n-1);
(79)
..24 ¢.JQ (Ag )
~=l
In fact, one gets equations
hence for
gt h,
(77).
= 0
(0<
= r;n;
g
= r;n::T).
"
(75)
From (77) and (76) one obtains (78).
develop from (77) and (78) when one takes h
=0
in (77).
The relations (79)
- 94·
In the following we sha.ll suppose that the constants C and C in (76) are
l
u~g)
chosen in such a way that
and
v~g)
:2 u~g\~g)
are normalized so that
= 1
(g = I,n-l).
~
Next we introduce the iterations
and consider
the obvious relations
(80)
which are identies for solutions
(73), (74)
and
(76).
Then one has the following theorem.
XL.
~
If
¢ is
~~decomposable and primitive and has distinct zeros
has identically
(81)
In fact, if one sets
l} ~~) = r 0
~_~
(0)
v~
+
r1
(1)
v~
+ ••• +
r n-l v (n-l)
~
-95.and if one multiplies this relation by u(;) one gets, by adding the results for
.I..
= T;n
u(g)and
.I..
and by taking into account the orthogonality and the normality of the
v(g)
.1..'
or, because of (80),
(g
= O,n-l),
hence (81).
Theorem D.
In the case considered
(82)
where
(83)
and
(84)
(g
= l,n-l).
In fact one easily obtains (82) from the fundamental relation
-96and by substituting in it the expression (81) for
cf!..~)
Relations (84) develop
4
immediately from (74) and (77).
B.
Let us still consider the case where the matrix
Aa
and primitive, but besides the simple zero
= 1,
¢ is non-decomposable
has multiple zeros
whose orders of multiplicity are
..., m •
IJ.
Then above all one has:
XLI..
... , '), ,
''"n
The rank ~ ~ determinant ¢(Ag )' Ag being one of the zeros AI' A ,
2
is not less than n - m
----
g
0
In fact, one sees it from the identity
¢(A) = ¢(a) +
A-a'"'
~ ¢~ I~ (a) +
~l
1 1
1
where a is a constant arbitrary number and
¢
~l
+ ••• ,
I1. (a),
¢
principal minors of ¢(a) of the orders n - 1, n - 2, ••• ,
Therefore the systems
admit of definite ana linearly independent solutions
(At-" = I:n) and u(g)
~
I
~1~2 ~1..(2
(a), ••• are the
-97in number I-Lg
~
m.
g
It can be supposed that these solutions are ortho.gonal and
normal bee ause if one had the solutions
which
~JOuld
not be orthogp nal and normal, but line arly independent, one would
always be able,
b;)r
a well known process, to replace them by their linear and
homogeneous combinations that represented orthogonal and normal solutions.
By using these solutions one will easily put f~) under the form
rO~~)
(B5)
'1 ""'t'
=
u~O) +
l"'
~
2 A(k)I/Ag)
g=l
g r..((3 ,
where
(86)
This result leads us to the following expression of
in the present
c~se:
(p~ = u~(0)) •
(d?)
We shall not consider the other cases of the matrix
00
Pkl~
¢,
because it is eas,y to
it according to the lines traced above.
13.
The preceding
Stabilized Chains.
n~~bers
can give us this general con-
elusion on the behavior of probabilities Pk/ ~ when number k varies.
where the matrix
modulus 1, the
In the case
¢ allows AO = 1 as a simple zero, and has no other zeros of the
PkJ~
have definite limits which are not dependent
on the initial
probabilities PQ..( and which represent the final probabilities of events '):3.
It is
the only case in which the limiting probabilities exist independent of the initial
probabilities,
because if
¢
has zeros of the modulus
1
others thaJll
Ao
1 or has ).,0 = 1 for a multiple zero,
=
Pkl~
or doe s not have a
determined limit, running through asymptotically and periodically definite positive values or has a determined limit but which is dependent
on the initial
probabilities.
Suppose that
¢ has
).,0
=1
for simple zero and that there are not other zeros
Then Pkl ~ have determined limit s
of modulus 1.
p~
that are independent of the
initial probabilities and there can be produced an interesting stochastic phenoi f one chases for the initial probabilities p~ of events A~ the final
menon:
probabilities
p~,
one 1<Till have a chain that we shall call stabi lized.
Its
characteristic property is that the probabilities Pk\ ~ for all the values of k
equal to
~ p~, ~d
~
preserve the
~
values for all the trials.
In fact, by taking the more general case B of the preceding number, one has
by (87),
by supposing
p~
But p
= p~.
~
..2;p
because
u(~)
and
v~)
~
, then,
by using (86),
IJ.g
-,
~
= u(O)
~
f~)
=-
2
i=l
are orthogonalo
(g)C1 u(O) v(g)
U~i
1:
;,t,.
~
==
0
(g
==
I;"S)
Consequently,
One see s then th at in the case of a stabilized chain each of the events
will have for all the trials a
sa~e
constant probability
p~o
~
However, one should
-99not confuse this chain of trials "lith trials \Jhen the probability of )
trial is a constant number
p~
in any
whatever may be the results of the other trials.
In fact, if one knows, for example, that in the initial trial we have stated event
ita/.. and if the results of all the other trials remain unknown, the probability of
~
th
in the k
test "rill be
a thing which is not equal to
p~.
Likewise, the probability of having in the initial trial· event Aa/.. and in
th
the k
trial event
~ under the condition that the results of the other trials
remain undetermined is
and the probability of having in the trials the numbers
the events
under the condition that one does not know the results of the other trials is
-lOO-
Chapter III.
14.
I40l1ENTS OF DISCRETE STOCHASTIC VARI.iliLES CONl'oJECTED "I'lITH SIIvIPLE
I'V.RKOFF CHAINS
Generalities~
Let x be a stochastic variable that can recieve finite
and well determined values
with the probabilities
respectively in the initial trial.
x
= x~
Let Px~(x~) be the probability of the equality
in any trial when one knows that x
= x~
in the preceding trial.
We shall
set
by supposing that f~ are constants for all the trials so that by considering the
values of x in the trials of the numerals 0, 1, 2, ••• , we shall obtain a simple
discrete I"1arkoff chain whose law is given by the matrix
¢ =
/~
( 11
ii2
o ••
{{~l
412
• ••
• •
0
• • • • • • •
•
• $
It is of advant age to consider with x the stochastic q uantitie suO' up u '
2
••• connected to numerical trials 0, 1, 2, ••• respectively and taking in the
corresponding trials the same values as x so that the probability of the equality
-101.
~+l
:::
x~
when one knoVls that uh
= x.,(
is precisely
f'13'
It can be said that the
quantities uo' u l ' u2 ' ••• are linked to the chain considered of the trials
characterized by the law
¢, or simply to chain W.
Following Markoff's notations let us still introduce the following quantities:
(88)
.,(k ::: E(u )
k
= mathematical
o'
calculated provided the values of u
A(i)
(89)
~
= E(uk+1.)
up • 00,
orovided
•
expectation of Uk'
~-l
remain undetermined, and
~K=. ,x( '
by supposing at the same time that the values of the other quantities
~
remained
undetermined.
Note some simple properties of a k and
XLII.
A(~).
One has
(90)
and
(91)
These relations are obvious.
The second shows us that
A5 i )
depends only on
.,( and i so that the nwrber k which by (89) enters into the definition of A(;) ,
can take on any value/)
XLIII.
(92)
We have the relation
-·102In fact by (90) and (91),
.,
~+i =
f
Pk+i
I ~x~
•
XLIV.
lie have
(93)
In fact,
~
f 4:/ ~t ?(p ~~-1) x~
- f q~ 'l
_-:"1)
.' (i-1)
il.
!~'
•
'l:Je shall consider further only the chains that we shall call normal by ineluding in this denomination chains whose laws are represented by non.. decomposable
and primitive matrices.
Then, since
¢ is the law of a normal chain, its
characteristic function is of the form
-103-
If..g I -<. 1
where
(g = l,j.l.).
We kno"r that a normal chain possesses limiting probabilities, well determined
and independent of the initial probabilities.
where
p~
is the final probability of the equality x =
and well determined quantities.
~ve
shall also write
(94)
by setting
s
"0
= .2i
(95)
g=l
Finally we shall write
s
Pkl~
(96)
by setting
(97)
XLV.
(98)
1V'e also know that for a normal chain
Let
a
=
= Pp
+
~ A.~ rp(f..g),
x~
and
¥'~:)
represent finite
-104-
Then, for
~
normal chain,
(99)
" A(i)
. = 1.~m
c(
lim a
"
J.7 00
whatever k
K+J.
=a
J.+CO
and.(~.
In fact,
~(;~ = PA
l.f t'
t'
l.L
~
+
g=l
then
.,.,
-'».2; P~X~ for i
--i'-
co ,
~
bec ause
IA.g ) "'" l( g = 1, l.L).
Next
'\+i
2~. Pkl~A(i) ~ ~
~
=
<I,
Pk/' a
'"
=a
for i-+ co.
XLVI.
\ve have
(100)
It is a simple consequence of the first of equalities (99) and of the following well known theorem:
~
lim
n-l>CI)
if
+ u2 +
m+l
n
u
lim n
n...,. 00 nm
II I
= A.
+ u
n
::.
A
m+l '
-10~·
Dispersion of ~ Sum of Quantities~. In the study of the sum of the
15.
quantities
'1l
or what amounts to the same thing, of the sum of the values of x in
the consecutivo . trials, the dispersion of this sum plays an important role and
we are going to examine it first of allo
For brevity let
of the expression
,
~ = ~ -
(u~
+
~
a.
+ •••
Let us envision the mathematic al expectation
u~_1)2,
by designating the number of trials by s.
We have
E(S~\:)2 = .5\ E(t\)2 + 2]/:(u' u;,).
h=O
h
g,h
One finds first
and next
E
'. '2)
(.6(u,)
h
11
=s
H01'18ver~ by (95),
jJ.
;:
-."
LJ
g=1
ep (g) ,
. .,(~
g
-106-
then, by taking into account (97),
--;-J
~ (xA- a)
~
2
<~
.~
r(h)
L1 PO"" L41..LA
""
h
-y
/"'
l_,\s
j.L
~
''I
= 2J --J ~
fI.
g=l
~
1-).,
g
(xA
- a)
t'
2
'fA (A.g ) •
t'
Consequently,
(101)
Let us now set
k and .e being any positive whole numbers.
s
kt
=:2
~ , if
=
(x-a)(Xv-a)
~
~7()
~
.,
1
(X't -a) p
2p
""
¥ -;;
0.
0
7.
A)(k) rn(.e)
l}/ ~ y
0"" l( ~
0
-> (x -a) (x "
<,( -a)
/. p
because
because
0
We shall have
[P
0""
~
+ ~(k)J p
""~
?{
-10'7On summing up Skt one will have
s-t-l
~
St::
~
Skt
k=O
=
Now,
=
IJ.
= (8-t)
PA +
l""
g=l
AS -. t
1
-
\[r(g),
g
r
1 _ ).
g
~~
1 _ ).s-t _
--g-lfA
().g ),
1 _ ).
l""
g
then
St
= ~ (x~-a)(xo-a)(s-t) p~-P(~)
~, t
.
IJ.
+
~ (x~-a)(xd' -a)'f~tj "2
~,~
g~l
1 _ A8 - t
_-:.:.g-LJ
I-A
.
g
~
(A).
g
s-l
Finally, by calculating the sum
,-,
~
St' we will obviously have the sum
t=l
~,
I
g,h
g
I
'~,
I
2
~ E(u ~), that goes back in to the expression E (~uh) •
-108We find
S 1
Ag - Ag +
"" [ ---.K...
SA
... .tiJ
g
J~ ~
- /(g)
(1 - A)2
g -
1 - Ag
0'
and
s-l
1 - A.g
1 - A.g
where
should be replaced by
1 - Ag
for g ::; h.
Consequently,
8-1
v
(102 ) ~
t=l
8
::;
t
,I r
~ E( u u )
g,h
g h
fJ.,
-", [
0::
~
g=l
sA.
--..Jl
l-A.g
-
A.
g
- A.
8+1]
g
(1-A )2
g
(x
~
- a)(x
""\ If )
-a) P ifF g
~
j3
~0
-109 ..
Thus we fi nd definitel y
(103)
• s
&J.
[
s~
2211_~
+
g=l
g
From this complete expression of
ai (2
u ) one finds immediately an asymptotic
h
and more simple expression:
(104)
Now let us consider the quantity
which is the dispersion of the sum
""
~
u h9
We have
E(
~
2J (uh - a»
2
""..,
- E (~( u h - a h
~ E [:2 ('1t-.) - 2: ('1t-'h) ]
y
= (..r;;j a h
- sa)
2
»2
[ :; (uh-.) +
:2 (uh-'h )]
-lJ.O.. ·
or
whence
(105)
But
and, as a result
~
~-,
" - sa = g=l
~ah
~
Ag
- AS
1
Ag
-
g
This quantity is evidently finite for all the values of s, then from the equality
(105) that can now be written
(106)
it is concluded that
(107)
Let us summarize our results.
-111-
where u ' a and a ~ quantities defined in paragraph 14, .Q!!£ ha.s,
h
h
2
,
and
•
16.
Moments of the
(108)
~roducts
z~
I
of u •
h
Let, for brevity,
= x~ - a, z;: x - a
and let us introduce the notations
(110)
jJ.
m
where m. and k. (i ;: 1,r)
1.
1.
;:
m
E (z ) ;:
":::'
~
P
.,1,..,1,.
(x
m
1
.....
-
a) ,
are posi ti va whole numbers.
-112(k k •• .kr )
We are going to derive a fundamental formula for the moments M l 2
~~ ...mr
which will be very useful in the following 9
(kl k 2 • ..kr )
XLVIII. The moments M m...
~ ~ calculated ~ ~ hele of ~
~ ~ •••mr
fallowing recurrent formula:
M(~k2" .kr )
~~ ...mr
(-r.- )
+
••.
m
••• z r
~r
...
~
z~
r
,
wnere all the sums should be taken for
---
----
a
= 1,
n;
~l
= 1,
n; ••• ;
~r =
1, n.
This formula is deduced very simply from the very defi ni tion of the moment con',idered.
One has
-lJJ-
m
r
.". Z.~r
m
r
••• zB
·r
+
(k1 k 2 ...kr 2)
,..
-·
M
~~
•••mr_2
11
~ PA
~r-1
kr )
13
mr-1
A
'r-1~r
ZA
mr
Z13
~r-1r
m
r
••• Zs
·r
=
etc.
-114By continuing in this manner, one obtains the formula (-l~).
All too additions
should be made for all the values of every Greek index that is found two or three
times under the sign of the sum.
By setting
(111)
•••
we can write the equality
(*) in the more condensed form:
1]
••• mr-
(mr )
+ •••
(-l~bis
)
In the fo1101.Qng the symbols [~m2
0"
m ] , (~m2.. ... m ) and (~m2 ... m ) I
t
t
t
will be called bracket parenthesis, and parenthesis prime respectively.
development (*bis) will be called development in parentheses.
The
-n5It is easy to see that the bracket [
of
(i~bis)
IrJ.IlL:2 •••
mr
Jis developed with tha help
into a sum of members each of which is a product of certain parentheses.
Let us take for example the brackets[2,3,4,5]"
-\
[2,3,4,5j
= [2,3,4J
[2,3,4J .. [2,3
We obtain successively
(5) + [2,3](4,5) + [2] (3,4,5) + (2,3,4,5) + (2,3,4,5)'
J (4)
+[21<3,4) + (2,3,4) + (2,3,4)'
[2,3]
= [2]
[2J
..
(2) + (2)',
=
(2)(3) + (2)'(3) + (2,3) + (2,3)1
..
(2)(3)(4) + (2)'(3)(4) + (2,3)(4) + (2,3)'(4)
+
(2)(3,4) + (2)'(3,4) + (2,3,4) + (2,3,4)1
=
(2)(3)(4)(5) + (2)'(3)(L)(5) + (2,3)(4)(5) + (2,3)'(4)(5)
(J) + (2,3) + (2,3) I
from which
[2,3]
[2,3,4]
and finally
[2,3,4,5J
+ (2)(3,4)(5) + (2)1(3,4)(5) + (2,3,4)(5) + (2,3,4)1(5)
+ (2)(3)(4,5) + (2)'(3)(4,5) + (2,3)(4,5) + (2,3)1(4,5)
+ (2)(3,4,5) + (2)'(3,4,5) + (2,3,4,5) + (2,3,4,5)1.
One sees that
[IrJ.~ •.•
m Jean b e developed into a sum of products of
r
parentheses by the following rule:
XLIX.
parentheses, it is necessary!:£. form
~
of
~
indices
order all the parentheses possible, and
~
[IrJ.'
In order to ~ ~ complete development of
~,~,
!:.9. ~
m2 , ••• , mr
J into
••• ,mr , without changing
the
~
of
~
products
parentheses by associating to every: product and parallel product. where
~ ~
parenthesis
~ ~
prime.
-116-
This rule is translated by the equality
t1!].~ •••
(112)
m
r
1 = :;q
(~ .. om
(
q1
)( mq ,+l •••m ) ••• (m +1·· .mr )
1
q2
qt
+ (1!]. ... m )I(m +l ... m ) ••• (m +1". mr)l ,
ql
q1
q2
qt
)
where the sum should be expanded to all the values of the indices q1,q2' ••• , qt
such that
(112')
It is easily verified that the number of the members of the development (112)
is equal to
O 1 + Cl
2(Crr-1 + ••• + Cr-l)
r _l
if all the m. are different from 1.
= 2r ,
But, if there are among the m. units, this
~
~
r
number is less than 2 because in this case all the parentheses (m.) disappear
~
identically where m
i
= 1,
(m ) beirg equal to
i
.::J p~. z~.
~.
~
~
~
= 1-\1 = 0
for m. = 1.
~
Now 1 et us consider t he more nts
(113)
-117One can calculate them either by the summation of the formula (*bis) or by
the summation of formula (112).
M
~~ ...mr
It is possible to get the general formula for
in this way; but the calculations are long and difficult and we shall
show only how one can do them and how one calculates the asymptotic value of
M
•
~m2··
.mr
Let us tackle the summation of (112).
\Je have
"<:l
k
~
l,
••• ,k
r
where A and A I have obvious meanings.
q
q
,~
A
~
kl •• .kr
q
One has then
=
(114)
by setting
k
and
(t)
= (k
qt
+2 k +3 ••• ,k ),
' qt'
r
designating by
'"
2J,
o
~
~
k'
the indices indicated b§ (114').
,
... ,
... the sums taken in relation to the
groups of
One obtains (114) by noting that the parentheses
(rn.. ... m ), (m l ••• m ), etc. do not depend upon the indices kl'kq +1 •••k +1
~
ql
q1+
q2
l'
qt
respectively (one sees this by (Ill» and that one can change tre order of the
-118-
Parallelly one has
where
... ,
(115 I)
and k I,
k(t) have the same Ireanings as in (114').
••• ,
Now let us see how one can carry out the summations indicated in
(114)~
By setting
k
50
= k1
=5
+ k2 + ••• + k ,
r
-
k + k + + k
+ ••• + k + k - t - 1
qt l
qt-l+1
ql 1
+ k
. . . . .. . . . .
5
t
:: S •
one finds first of all
...<il=~
and then
kl=O
• t,
.....
..
k + k
+ 1,
qt+l
k(t+l)
q1
...
=5
-
k + k(t+l). t - 1,
-119-
. . . . . . . . . .. . . . . .
sl(8 +1) ••• (sl+t -l)
1
1.2 ••• t
"
. . . .. . . . . .
sO(sO+l) ••• (sO+t)
1.2 •••t(t+l)
=
,
then
sO(SO+l) ••• (sO+t)
(t + 1)1
(116)
(s_k+k(t+l) -1) ••• (s_k+k(t+l)_t_l)
=
(t + 1)!
•
=
t+l
CSl
,
= S -
Sl
k + k
(t+l)
- 1 ,
(117)
Similarly one finds
s" = s - k +
~(t+l) - 1;
•••
Now let us consider (117).
It is necessary above everything else to calculate
the last sum of the right part of this equality.
2:
k(t)
t 1
C + (m
Sl
qt
\l1e find
+1 ••• m )
r
ct +l
Sl
m
... z~rr
-J20i-
... -p ~r
(k )
(119)
•••
~
,
r-l r
- (k )
... Ij[ ~rr-l ~r
=
(120)
k
~
qt+2
f1r(kqt+2)
"'"7"1
•••
~qt+l~qt+2
.d.I
i
k ., .
r-.L
kr - l )
~r-2~r-l
t l 1J1" (kr )
2J Cs +I
-,;--.
k
r
~r-l~r
J
where the la st sums should be calculated for
k
k
(121)
k
r-l
r
=
1,2, ••• , s - k + k
= 1,2,
"'J s - k + k
(t+l)
(t+l)
+ kr - t - 2,
+ kr + kr- 1 - t - 3 )
. . . . .... . . . . . . . . . . . . . . . .. . . . .
qt+2
= 1,2, ••. , s - k
.
T
k
(t+l)
+ k
(t)
- t - r + qt+2 - 1
and are calculated wi. th the help of the known formula of the theory of finite
differences:
(122)
where ~(n) is a polynomial of the degree m in n and the undefined sum and the
differences are taken in relation to n.
By setting in (122)
~121-
by next applying (122) to the last sum of (120) and by taking into account (121),
we will have
'~ t+llTr(kr )
~C ,
k
r
s
!'
k
A r Ct +1
g
s'
R
t:l
t'r-F'r
""
I.l.
v~g)
~ t:l
A
g=l I-'r-1t'r
I
k
r
,.A
......
g~
/..- - 1
g
+
C
rct+l
L s'
A
g
A - 1
g
l\C t +1 +
s'
•••
k =s_k+k(t+l)+k -t-1
r
r
k =1
r
But
ct+1 =
SI(SI -
Sl
l\C t +1
s'
...
1) ••• (s' - t)
(t+1)f
•
(s '-1)( s 1-2 ) ••• (Sl_t)
= -
t
f
,
... ..
•
.......
and all the quantities, with the exception of the last, are reduced to zero for
k = s - k + k(t+1)+ k - t - 1 = Sl - t + k , then
r
r
r
I
g
1>
Ag - 1
[ Ct +1 •
s'
A
(123)
= -
g
Ag - 1
A
~
9
A - 1
g
[c
l\C t+l
s I + •••
t +1 + Ag
sI
A - 1
g
J
t
Cs '_l
k =s-k+k
+ C r
k =1
r
+
(t+1)
oj.
•.• +(1)1 ) t
kr -t-l
J
1
Cs'-t k =1
r
-12lcr
(124)
=s
- k + k
r
+ k(t+l) - 2,
or again to simplify the writing
(125)
by setting
K (
(126)
).) =
s, g
[C t +l +
t..g
1 - t..
cr
t..
- 1
g
g
Ag
C~_l+
v
t
g
•
U
+(t.. t..-1)
g
l
Ccr-t·
]
K( s, A ) is a rational functi on in 't.. and a polynomial of the degree t + 1 '\<d. th
g
g
relation to s; the indices k ,
l
~, ••• only enter there by C~+l, C;_l' ••• ;
let us note also that
't..
s
K (s, A.)I'\,
g
g
1 - A.g
(127)
t+l
(t +
lJr•
Consequently,
(128)
,',
~
k
r
t+l
C ,
S
- (kr)
Y! ~r-l~r
Q
=
y.
~ K(s, t.. )
g=l
g
r
( )
IJ) g
~r-l~r
then, by (120),
,
'
But
(129)
and by using again the formula (122)" one will see that
k
~ k 1
~ II r- K( s,)" )
r-l
gl
g
= K( s, A ,A ),
g
gl
where K(s, A ,A) is a rational function of A and A and a polynomial of the
g gl
g
gl
degree t + 1 with relation to e; ita asymptotic valUG is givon by the equality
t+l
s
(t+l)ta
(130)
by continuing in this manner one finally arri ves at the result:
... ,
-124where the sum should be taken for
•• c:,•
g = l,~;
and K(s,'A. ,). , ... , 'A.
) is a function parallel to functions K(s,'A. ),
g gl
gr-q -1
g
K(s, 'A. , 'A. ), ... and ha viri f or their asymptotical value
g gl
s
t+l
(t + 1) r
•
By using this result we have through (119),
'>
.ti.d
Ct +I l
k(t) s
m
( m +1. om )
0
qt
r
=
••• z r
~r
(131)
where L(s,~,~,
.0.,
h~) is a rational function of ~,'A.2' ••• , 'A.~ and a polynomial
of the degree t + 1 with
t+l
s
(t+l) ,
•
.0 .
~
having for an asymptotic value
h A ... A
gl_ _ _gr-q
-1
_ _ _g_
_--=t:-
• •• (I-A
)
gr-qt-l
(132)
•••
•
-125 -
By corning back to the equality (117) one now see s that one can write
By the method described just now one will calculate the sum
"'"
Li:J
and so forth.
k(t-l)
The final result will be that the sum considered of A
q
\J,
rational function of AI' 11.2 ' ••• ,
is a
and a polynomial of the degree t + 1 vnth s
having for its asymptotic value
s
(133)
t+l
...
(t+l)~
1
in which lJ.".,
tll- .• m
by (132).
, • eo have meanings absolutely parallel to that which is defined
ql
The same line of reasoning is applied to the sum (118) and shows us that it
is a polynomial of the degree t in s.
Let
I
be values of the sums of A and A that are obtained by the method described.
q
q
-126Then
(134)
where the sum should be taken in (112).
This formula leads us to an important conclusion.
Let T be the maximal value of t + 1 that is definitely attai ned for a few
members of the development
(134).
Then evidently
or, by taking into account wha t was said a bout the asymptotic values of the sums
of A ,
q
r
(135)
•
• • • lJ.
m
+1' om
qT-l
r
, which give us the asymptotic value of M
The members K
lrJ.~ ... mr
qlq2" "QT-l
and which will be called principal members of M
, are obtained
~~ ...mr
by the summation with relation to k ,k , ••• , k of members of the development
l 2
r
of the bracket [~~ ...mrJ into parentheses which have the greatest possible
number of T of the parentheses factors (~ ...m ), (m +l ...m ), ... which are
Ql
Ql
Q2 •
not reduced to zero indentically, that is, which do not contain among them one
or several parentheses (1)
of the development of
or again orde!. of
= ~ = O.
[lrJ.m2 •••mr
[tn:I.~ ••• mrJ
•
J
We shall call such numbers principal numbers
into parentheses and the number T their order
-127One can easily determine in each particular case the principal members of
[~m2 •••mr ]
and then wi th the help of (13.5) the asymptotic value of M
~~
•••mr
•
For example, by retaining only the pri.ncipa1 members in tm
developments one has
[.5, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 2] = (5)(3)(1,1)(1,7)(1,1)(1,1)(2)
+(.5)(3,1)(1,1)(7)(1,1)(1,1)(2)+ ••• ,T=7·,
[.5, 3, 2, 7,
3J = (.5)(3)(2)(7)(3) + ••• ,
T = .5;
[1, 1, 1, 1, 3, 5] = (1,1)(1,1)(3)(5) + ••• ,
[1, 1, 1, 5, 3J
One obtains from this
=
T
= 4;
(1)(1,1)(5)(3) + (1,1,1)(5)(3) + ••• ,
T
=
3.
- J:28-
where
IJ.
~
I
IJ.:l.
,1
=~
~
I
~,l,l
etc.
1 - A.g
g=l
=
k..
-~,
k_ k
-"'2, 3
It should be noted that in the last sums the limits between which one should
add with relation to
lJ.
for
~
or to
kt ,
indices of the corresponding moment M
k2.
1
~3'
~m2 oo·VIr
s-5
IJ.
=}:
•
Thus, for our examples,
( )
::£ (Ir kl
k =0
1
for 1J.i,1,1 depend upon the number of
1'.,(~
=
1 _ As - 4
g
g=l
I
1J.:l.,1,1
17.
l'1oments of the
~
I
of uho
Limit Theorem of p,robabilities of
Let us know show the following important theorem.
these~.
-129 -
- - -
We have the .;......;;~.-.-.;~
relations
LI.
(136)
where
(137 )
and
s-l
E(
(138)
~Uh
) 2
m
+1
= O(sm).
h=O
The demonstration of the quality (136) rests on the identity
s-1
( ~'
h~O'1l
)
m
) r,
*•• + kr
where the sum is taken for all the whole and positive values of the in dices
m and k such that
i
1
r
(139)
~s;
One can still write
(2m) ~
o
•
• m1
r
• • • (u~
1
m
+.u + k ) r
r
'
-.130-
whence
• • • mr
2
In order to have (136), it is necessary then to ShOH the relation
Let us take for thi s purpose the bracket [mrm2 • • • m
-
that its largest order T .. m if m. satisfy (139).
l.
r-
7 and
let us show
In other terms, let us show
that the largest number T of the parentheses factors that can be formed from the
following
as equal to m when m. vary in every conceivable manner.
J.
I say that this number T = m is reached for the bracket
L2, 2, •• 0' 2_7
(m times the index 2) and that these brackets that are obtained by replacing
in every manner possible the 2 1 s by 1, 1, that is, the brackets such that for
example,
(a)
L2,1,1,2,2,1,1,1,1,2,2,
0
•
0, 2.7 ~ Li,1,2,1,1,1,1,2,2, •
0
•
2,1,1.7, ••• ,
-131·
and that the order T of the other brackets, having ~ m. = 2m, is less than m.
J.
For brevity let us call
L2,2, ••
~,
2.7
and the brackets of the form
(143) the principal brackets of the sum (141).
One sees
T
= m and
f~rst
of all that the order of the principal brackets is in fact
that each principal bracket, developed into parentheses, has only a
single member composed of parentheses (2) and (1,1) so
that~
for example, for
the brackets (.a) the principal members are
(2)(1,1)(2)(2)(1,1)(1,1)(2)(2) ••
0
(2)
and
(1,1)(2)(1,1)(1,1)(2)(2) ••• (2)(1,1)0
By trying other compositions, for example, for the first bracket,
(2,1)(1,2)(2)(1,1)(1,1)(2)(2)
•
0
• (2),
one will always have an increase in the number of parentheses consisting of two
indices and a decrease that consists of one index, then a decrease of the order
of the member considered o
Let us next see that the orders of the non principal brackets of the sum
(141) are"" m.
The non principal bracke ts are obtained from the principal brackets or even
from a single one of them, from the bracket
(2 m times 1)
-1)2."
by replacing the different groups of units by their sum so that one will have
brackets that consist either of indices 2 ar..d 1, the unit being found in groups of
odd number, even or of odd indices, the even indices being ~ 4 or t.he odd indices
being
m.
~
3
g
The order of the bracket of t the last class is obviously less than
As to the order of the brackets that contain only the indices 2 and 1, unity
or the unit being found in odd number group, one notes that these odd groups of
units should be themselves of even number.
For exarrp1e, one should have
~l, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, ••• ,
2_7.
Now, the two groups 1, 1, 1 and 1, 1, 1, 1, 1 and the group 2, 2 that is placed
between them, gives the greatest number of parenthesis factors (1$1)(1,2)(2,1)
(1,1)(1,1) for example, equal to
5 that
is less than
6, the greatest number of
p:lrenthesis factors that can be formed by the group for example,
1, 1, 1, 1, 1, 1, 1, 1, 2, 2,
consisting of the same indices, but ha vlng units assembled in even groups.
This
remark is general and one sees that the non principal brackets considered also
have the orders <m o
Then, the orders of all the non principal brackets are" m.
Consequently,
in order to have the asymptotic value of the sum (141), it is necessary to consider there only the morne nts 1'1
~m2" .mr
the principal brackets.
vJe are
nOvi
tha t are obtai ne d from the summati on of
goi!1f-; to calculate the sums of these brackets.
-133 First consider the pri.ncipal brackets that consist of 2-t units and of
m-t secorrl. Obviously there are C~ of them.
Let C be one of them and
ti
(i = 1, 2, ••• , ct;
r = m + t).
m
(142)
The principal member of C is a product of m - t parentheses (2) and the t
ti
parentheses (1,1) and its order is
'T"1
~
k
k
1- - - r
following (133).
sm
m;
then
m-t ( t ) t
u...
'L,l
C,.-".1
I ' lJ.2
v~ m.
c
sm
m•12$
But (133) was obtained without a detailed calculation and since
the relation (143) plays a very important role further on, we smll show it
separately _
Let us take for greater simplicity a particular bracket,
(144)
f or example, and calculate the a symptoti c value of the sum M-"2, '...l , 1 , 1 , 1 , 1 , 1 , 1·
The principal Irember of (114) is
-1:4-
then
]"L
-~,2,1,1,1,1,1,1
-'V IJ,
2
.2
(145)
~
~
1"
k1 ,k2, k)
kS,k1
where one should add for
k8
= 1,2,
... , s - k4
- k6
- 6
=s8
-135We now have \see (116)
8
sl
~I
~
21 1= ~
k1 ,k 2 .k3,
k1=O
k
~
k =1
3
5, k 7
=
3
"'0
(s-k4-k6-ka-1)(s-k4-k6-ka-2) ••• (S-k4-k6 -k -5)
a
5'•
= c5s-k
lI. -k6-k8 -1
= c5Sl'
s'
=s
- k
4 - k6 -
k ~n
-
1,
then
=
But, by using equality (122) one finds
5 5 5 .]
~i " "
-A
• •• +(' A-
~
=
where s"
=s
- k
4 - k6 -
2.
Then
Cs '
Ie =5 +1
+C. 8 B
k
=1
B
-136...
and
= ::;
p P
7, 8
~
p
z
~7 B7 ~8
IJ.
~
rv C5 ~
s" gel
C5
I
s" 1J.1,1
=
z
t..
-L
1-),
g
= 1-2
2
g=l
K(s,t.. ) JJ(g)
g f ~7~8
~ p r(g) z z
B
. 7,'B8 P7 ~7~8 ~7 ~8
5
Cs II B.
Consequently,
1
rv - B
2
=
1
2'
5
B CSll
'
'
_1 5
4" CSIl '
~,l -
B2
,
SIll
= S - k
4
-
30
-137Finally, absolutely parallelly,one finds that the asymptotic value of the multiple
sum of the right part of (145) is equal to
from which
(147)
as should be obtained by (143).
It is not difficult to apply the method explained to any pr'incipal bracket
and to verify the formula (143) in general.
= 1,
The relation (143) is valid for i
This asymptotic equality takes place for
t
2, ••• , C and (142) gives
m
t = 0,
1, 2, ••• , m, a fact which is
seen immediately from the demonstration of equality (147).
The quantity S,e enters into E
~ut)2m
I-lith the coefficient
But
t .1
= Cm
.3
c ••• ( 2m - 1 ) s
m A
m-t Be,
.~
-138-
then
2m
E
(:~:u~) ~ 1
• 3 •
5 ...
(2m - 1) sm
the re1a tion (136) is found proven ~
The equal i ty
(138)
is proved now wi thout difficulty.
In fact, in order to see tha t it is true it
is necessary only to note that the brackets
(148)
are obtained from brackets
(149)
~
+ ~ + ••• + mr = 2m,
considered above, either by intercalation of the supplementary index 1:; or by
increasing one of the indicesby 1, and in all these cases the order of the transformed bracket (149) cannot be increased.
is ~ m, from which (138) results ..
Then the order of the brackets (148)
-139The relations (136) and (138) allow us to prove very simply the following
s-l
~
theorem on the limit of probabilities of the sums for
for s -700:
I
JCJ u h
h=O
Theorem E.
For Markoff chains of which the law
¢ is non decomposable and
primitive and admits >"0 = 1 as a simple zero and has no other zeros of the
modulus 1, the probability of the inequalities
for
s~ <D
tend uniformly toward t.he limit
J
1 t2
{3
1
'\/Tn
e-
2
dt
~
whatever may be the real numbers .t.. and ~, provided one has ~
:f
O.
This theorem is a simple consequence of the well known theorem of TchebycheffMarkoff
and of the relations
The relation
(104)
(136), (138),
and
(104).
is equivalent to the equality
~rvs (A + B),
which is besides a special case of (136).
M2 :f
Then from (136) for
0, one finds
2m
lim E
s~oo
(
U
o
+
~
+ ••• + u 5-1
I\!M;
- ~)= 1
• 3 • 5 • •• (2m - 1)
_;h t 2
00
1
=1~2n
J"
-00
t
2m
e
2
dt,
and from (138)
+ u s _1 - sa) 2m+l • 0
lim E
87 en
f
QO
c:
1
,,(21i
2
t 2m+l e -t dt •
-co
Consequently, by virtue of Tshebysceff-Markoff's theorem we have:
~
(150)
+ ••• + u _l-sa
s
uniformly for all real values of
~
and
B)
<~
c:
--=.1_
J
dt
;"J2; ""
~o
By recalling the connection between the quantities u
and the stocmstic
h
variable x considered in,p~ 14, one can identify the sum uo+u + ••• +u _
s l
l
wi th the sum of the values that x takes on in tht.: numerical trials
0,1,2, ••• , s-l. By designating this sum by Xo + Xl + ••• + x s _l one can still
write (150) as follows:
lim P
s7- co
•
(.;...~-<.x0
+
~
x
+ ••• + s _l - sa
< 131~
--Z
..
dt.