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BINARY RANDOM WALK ON A SEMI-INFINITE LINE WITH ONLY
AN UPPER VARIABIE ABSORBmG BARRIER
by
G. B. Tranquilli
University of North Carolina and
University of Rome
!nstitute of Statistics M1meo Series No. 413
November
1964
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This research was supported by Air Force Office
of Scientific Research Grant No. 84-63.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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BmARY RANDOM WALK ON A SEMI-INFINITE LINE WITH cm,y
AN UPPER VARIABU: ABSORBING BARRIER.
by
G. B. Tranquilli
Uni versity of North Carolina and
University of Rome
-----------------------------------------------------------------------------Some statistical problems may be described by a binary randall walk
scheme an a semi-infinite line limited by an upper absorbing barrier variable
with the number
n
of steps.
If the absorbing barrier changes linearly known synthetical. methods mB\Y'
be applied directly givmg the exact value (or optimal approximation) of the
probability of absorption P and average number
absorption (if pal).
ii of steps leading to the
These results allow to give an extensive sufficient
condi tion to be P=l when the a})sorbing barrier is not changing linearly; in
this case the problem of the distribution of the number
to the absorption MUst
be investigated
n
of steps leading
to estimate P (if P < 1) or ii
(if P=l).
Any binary random waJ..k scheme can be transformd into an equivalent
scheme whose binary system of displacements at each step of the random walk
is (0,1).
By using this system of displacements, if the oorrespanding absorb-
ing barrier is not decreasing with
tion probability Pen)
after
probabilities P(m), m < n.
n
n
a recurrent expression of the absorp-
steps exists as function of scm! preceding
I
C~
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I (General Preliminaries)
I.l - There are statistical problems which:
i)
consider sequences (ON) of independent trials, or steps, or
staees) or experiments (in general operations
°)
in each of which one of
n
necessary and incompatible quantitative events, either
If x
(On
n
Xl
t~
or x", happens.
is the n. th value of the sequence (x)
n
-+
associated to the sequence (O )
n
( 0)
xn ), it is a realization of the r. v.
"xn" = "Jet, (independent of n)
which assumes the V'&J.ue XnctX. I with probability q and the value x n- x" with
probability p (x' < x", p+q=l) ;
ii)
are concerned with the sequence (x(n»
of the partial series
n
aS80ciated to the arrangement of x' and x" in a
i=l
same sequence (~'X2I .••
x(n) =
1: Xi
'ZnI •••• )
iii)
define
n
1) for each n an event En c: R which happens when for the first time the
sequence
i < n) assumes
boundary A(n)
(1)
exact~
(x(i»
&
at the n. th operation On
(and not before for any
value x(n) = l'(n) Which is equal or greater of a given
> nx' depernUng on
(~,~, ••• ,Xn)E En
n:
/_.... {X(i) < A(i)
~
x(n)
=x~(n)
for each i=l,2, ••• ,n-l
~A(n)
co
2) the total event E = U En
n=l
3) the event E cootrary to E:
(X2)~, ••• ,xn, ••• )E
i <==> x(n) <A(n) for each n=1,2, ••• ,co ~i c: R~ ;
(By definition the events E , n=l, 2, ••• ,
n
co ,
and i are necessary and in-
compatible: some one of these events can be empty);
(0)
We will indicate the r.v.
IS
bet_en inverted
2
COJJlD&S
n(r.v)".
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iv)
stop the sequence (0 )
m
at the n. th operation 0
E occurs;
n
n
if the event
00
aim at estimating the probability P=pr(E) = 1: preEn) that an
n=l
event En € E will occur for any finite n: pr(E U i) = pr(E) + pr(E') = 1;
v)
vi)
define the r.v. "n"
which assume the va:llles n=1~2, ... ,00 with
probability
pen)
00
= pr("n,.-n) = pr(E)
n
P = n~ pen) ~ 1
:+ a~oo pen)
for each finite n (necessarily
= 0 ),
* = pr("nll=oo)=pr(E) = 1 - P
p(oo)
(if P
for n =
00
< 1 the r.v. "n" is degenerate);
00
vii)
operations
r. v. fInn
P
aim to an estimation of the average nuni:>er
0
n
which do hlq)pen the event
1: nP(n) of
n=l
E, if pel (only :in tbi s case the
is not degez;fate and. the average number
< 1 -> Ii =
Ii =
n
has rll:!aning:
00 ) ;
viii) aim to an estimation of the probabillty P=pr(E) < 1 when the
event E
is delimited to the set of all the Enfor n
the sequence (0 )
n
~!!
by a truncation of
did rot lead to E for
n
n < n. Each E
will be empty for n > n : E=E( ) = U E ;
-n
.!!
n=l n
(2):'" .,Xn ) € i=i(n) <=> x(n) < A(n) for each n=l, .. ,!!.
-
1.2-
at the..n.. th "operation" if (0 )
n
-
Such problems may be described by a binary scheme of r. w. (random waJ.k)
on a semi-infinite line [xl, 'With the a. b. (absorbing barrier) defiDed by a
sequence
(A(n) } (n=l,2, ... ) as ~r bound variable with the number n of
steps: (x t ,x") is the binary system of displacements at each n.th step
[pre ,tXn,,=:=X' )=q, pre IIX l=X")=P, pr(xo =0 )=1 ].
n
I
n
Since x(n) = 1: xi:s nx" if x(m) < A (m) for every m=l,2, ••• ,n we have
i=l
x(n) .. (n-m)xll :s x(m) < A(m) ==> x(n) < min [A(n); A(m)+(n-m)x ll ; nx"}
m<n
Then the r.w. may assume on the semi-infinite line (x] the positions
x(n) uith
nx':S x(n) ~ i(n)
(2)
III
max [x(n) : x(n) < min
(A(n); A(m)+(n-m)x" ;nx"}] •
m<n
~'lith regard to the sequence
(i(n)} we find
max x(n-l)+x"=i(n-l) + x" (xn=x"), or
x(n)
= x(n-l)+xn ==> max x(n) =x(n) =1:
max
x(n-l)+x'~(n-l)+x'
x(n-l)+~
<
(xn=x'), or
x (n-l)+X'
from wnich
x(n) -'i(n-l) ~ x' or = x"
Re~mbering
for each
---> ''E
n
(x' < x ft )
•
that in any one sequence (x(m)} we have x(lll)=x(m-l)+x
m(x(O)-o), then
non-empty
if He suppose x(n)=x
(n)=x(n-l)+x
ri
,,(x A=x ) necessararily we must have tile reversible
n
n
implications
3
(:~'(!ll»
such that i) x'(m-l)+x =x'(m) < x(m)<A(m)
m
-
ii) x' (n-l)=x(n-l)
<===> x(n-l)+x; ~
for m:l;.2 •• n-l
,
at this general result
x" if 'X(n)-x(n-l)=X'
x(n)-x(n-l) < x' <==-> E non -empt y' r'=
n
( n { x' or x" if x(n)-x(n-l)
Hence follo,"Ting (3) we arrive
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x(n)-x(n-l) = x" <=-'> En empty •
Moreover
(5)
E non-empty <::--> i(n) < :<:(n)=x"(n)=x(n-l)+x" < x(n-l)+x" < x(n-l)+x ll
n
n==> x"(n)~x" :s x(n-l) < A (n)
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,
x(n-l)+x~ = x(n)=x A (n) ~ A(n) > i(n)
< :: > x (nhx(n-l) < ~ .
(4)
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=x~
<==--:>
n n
,x
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Tile events
(6)
{
n
0
n+
E
n
are
= min(n
= min(n
The integers
no
extremit;}" of the r.v. "n"
empty also for n<n
cert,;ainl~y
and n>n+
0
where
n::e" ~ A(n»)
nx' ~ A(n») (n+ can be inexistent: neither finite nor
infinite)
and
n+
are
respectivel~r
"I'lhose range is then
Necessary and sufficient condition for
the lower and the upper
n -n
+
n
o
<
0
co
•
that is in order far
the r. "IT. to be absorbed at least once after a finite number n of steps, is
A(n)
.:s nx"
for some finite n.
~le
non-existence of
n (or n = co
o
0
)
is the
neceSSlU"'J and sufficient condition for P=O.
Necessary and sufficient condition for n+
> 1 that is
in order for
r.il. to assume at least one llOsition Y'J.=x'
before absorption,is A(l) >x';
other.rise the scheme't'lould be meaningless.
The existence of
infinite .' by implyine; the emptiness of the event
E,
n+ finite or
is a sufficient cornlition
for P=l.
But this conditicn is too restrictive to imply
P=l.
1ie
shall give as
conclusion of Chap.II a wider sufficient condition about the behavior of the
sequence (A(n)) as n
-.co
conditior~ pr(E)=O even if
in order far E to imp~' P=pr(E)=l: under that
E is not empty (n+ inexistent).
In general from the a.bove-mentioned problems one did not require the
distribution of the r.v. number "n"
of steps leading either to absorption
(event E) or to no absorption (event E), from which it 't'Tould be possible dra."I>1
directlJ.- and exactly the probability P=pr(E), as well as the other pertinent
indexes (n=E(nn,,) / Var ("n,,), •••• ) if P=l.
In fact a metl10d for an approxi-
mate or direct calculation of ~rf-pr(En) has not been until nO"!'T investigated
I.
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in nell of the complexities of solutions which besides are not considered
absolutely necessary to find P=pr(E) or an its approximation.
5
This because
I
general~
simpler schemes have been.. until now, considered based on linearly
variabl e a. b. 's: A{n)-Sn+A uith S ~ x' J A >
o.
In such a case we sllall show (Chap. II) that is indeed possible to
app~r
known synthetical methods which give good estimtion of the statistical
Ii if P=l),
features of the considered linear scheme of r.w. (P J and eventually
according to the Wald' s schere for the Sequential Analysis of a Binomial
Distribution.
If hO'\'1ever the a. b. f,A{n)}
is not linear the. just now mentioned
synthetical methods can be applied to obtain
i)
an estimatioo of P (if P
< 1) and n (if P=l) by bounding
(A{n)} above and below by linear a. b. 's;
ii) an extensive suffi cient condi t1 on abCllt the behavior of the
sequence
(A{n»
as n
~ 00
in order tor the absorption probability to be
P=l.
However mostly such estimations ot P{if P < 1) a:ld
coarse to be utilizable.
ii (if P=l) are too
What is left is to find a direct wa:y yielding an
expression, a method of calculation of the probability P{n) = pr{E ) for
n
each n.
n'
Then the detective estimations P' III 1: P{n) of P (if P < 1) and
n'
n=l
=
t nP{n) + (n '+l){l-P') of ii (it P=l) will be nore precise if a
n=l
larger number n' ot probabilities P{n) (n=1)2 J • • • J n') are calculated.
n'
The description ot such a method is the specific object of the
present work in which {Chap.
III} we shall investigate specially the case
of nonincreasing or nondecreasing sequences
(a{n}) =(
A{~)-~'
x -x
)
•
The extension of the r.w. on a semi-infinite line bounded only abave
by an a. b. to the case of a. b. 's not linearly variable may satisfy better
some problems which are solved only approximately from the schemes of r. w.
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witll linear a.b.
At the same time it may lead to the formulation of new
JOOre general problems which \'lOuld be an extension of
SCllE
}~o"m
ones.
n
1.3 - Let us indicate by flR.H. [(-l,+l); (q,p); -b < 1: z. < a»)" the
i=l J.
classical scheme of unidimensional r. w. on the z-axis '\-lith t'tol:> a. b. 's defined
by (-b) and (a) (ab
n. til step
> l) a>
by the r. v. "zn'"
0., b
> 0).
Indicating the displacenent at the
onl;y two values are possible:
z
n
= -1
with
probabilitya=l-p, and zn=l with probability p independent of n; (-b) and (a)
are respectively the lower and upper boundaries of the open interval on 'mi Cll
the r.11. is walking starting from the position z=O.
of tile r. vI.
the a. b.
I
The possible positions
before crossing With the a. b. and ending ..lith the absorption b".f
are the integers
n
I'
Zi
< a,
n=l,2" ••••.
i=
Since in the following speaking of r. w. we will mean an unidimensional
binomial r.w. (only two displacements necessary and incompatible at each
step) with only the upper boundary as a. b., and since we 'tilll agree to
attribute always the probability p to the greater vallle (the second one) of
the tHO displacements, we can vlrite for convenience
R.H.
[ (-l,l);(q,P);-co <
The general extension of
~
i=l
SUCi]
z. < a) ]
IE
R.W. [ (-1,1);
J.
~
zi < a) ] .
i=l
a scheme is given by a
R. H. [(x', x") ;x(n) < A{n») "those two displacements of probability q and p are
at each n.th step respectively x =x',)c =x" with x' < x", and 'Whose possible
n n n
positions are the values x{n) =
1: Xi < A(n), where the sequence (A(n)}
1=
is an upper variable a. b.
He then observe that if fr01-:l a system of displacemcnts (x' ,x") we pass
7
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to the
system (z', z")=(ux'+v,
this is equivalent to make the
"x"
n
-+ "z "
n
=u
~,,<:"+v)
with invariant proba.bilities (q) p),
follo~dng
llneer transformation of the r.v.
"xn ,,+v for each n, and this inq>lies
z(n)
= ux(n)+vn.
Since we want to maintain the inequality x" > x' , '\ore nmst have u > 0
as can be seen from the relation
x' < x", z '=ux'+v < zIt = ux"+v
(C(n»
If
U
=
zit -z •
X " -x f
is the a. b. of the R. W. [(z', z") ;x(n)
> 0,
V
=
Z • -x I Z It
X " -x i
< C(n)], this r. w.
scheme is equivalent to the preceding R. W. [(x' ,x") ;x(n) < A(n)] if
C(n)= uA(n)+vn; in fact we have the relation
(8,i)
=
x(n)
< A(n) ~ z(n)=u x(n)+vn < uA(n)+vn
z(n)- vn
When x(n)
n
~
A(n) for tl'l!! first time, necessarily z(n)
~
=C(n).
C(n) for the
first time:
x(n) = x A (n) ~ A(n)
(8,ii)
So
~'le
<==> z(n) = zA(n) =
UXA
(n)
+Yo ~
C(n)
obtain the folloldng relation of equivalence annng r.w. schema:
R.W. [(x',x"); x(n)
< A(n)] S
R.W. [(Z',Z"); z(n)
< C(n)] ,
where, from (7) and (8,i)
i) C(n)-uA(n)+'Vn
ii) (z I, zit}:::>
-=>/::.::. :
z(n)= ux(n)+vn
z"-z I
x"z I_X ' Z " n
(n)=
x' x(n)+ x" x'
- -.
t
C(n)=
iii) A( 1)
v"
Z:'-Zl
Xii_X'
A(n)
given
x· ,x"),A(n)
X"ZI-X'Z It n
Xii_X'
> x' -=>uA(l) > UX'.Z'_V <=> C(l)=uA(l)+v > ux'+v
(in order for the r.w. to exist).
8
= z·
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From (4) "ire obtain another relation of equivalence.
consic'..er tile class [Ao//(A(n)}] of the a.b. 's
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{
tlr.t
that x(n) < x(n-l)+x' «==> E non-empty)
n
b) i(n) < AO(n) for every n SUCil that
x(n)=x(n-l)+x ll ( <=> En emptyL
the sequence of events (E ) does not change when we consider
n
(AG(n)}
an~r
a. b.
Then this class is an cql.1ivalent class
of the class (Ao// (A(n»].
with reGard to the equivalence relation which establishes the same sequence
(E)
n
(the class includes the a.b. (A(n))
class ilas no lower extreme).
which is its representative; the
Hence
R.H. [(x', x");x(n) < A(n)] ==
R.vl. [(x',x");x(n) < AO(n)] for every
Tile analysis of every R.I'I. [(x' ,x"); x(n) < A(n)] "r.i.ll be carried out
more conveniently by considering tlle equivalent scheme R. H. [(0,1) ;f(n) < a(n)]
"There, from (9,ii) (Z'=O, z"=l, C(n)=a(n), z (n)=f(n», we have
i)
(11)
5.i)
iii)
a(n)= A(n)-ox'
x"-x'
x(n) ox'
~;
f(n)=
x -x
i(n) = (n-f(n»
> 0 ->x(n) = (n-f(n»
-
Xl
x' + f(n)x lt
+ f(n)x".
From (11, ii) we deduce fen) is the frequency of the c.isplacements
equal to x" af'ter
fen) =
(12)
{
I.
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SUCd
a) i(n) < AO(n) < i(n)+(x"-x') for every n sucll
aI
I
(AO(n)}
In fact if we
So (2)}
~
n
steps;
t~1at
is
f. , where f.=
~
~(n)-~(:::) ~
=
(4), (5), (6)
n
~
x.~ -x'
f'=O
,'lith probability q=l-p
-x'
f"=l
with probability p
X"
= 0,1 •
and (10) Cive
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i)
ii)
0 ~ fen)
no
= min
~
fen)
(n:a(n)
= max
~
[fen): fen) < min (a(n); a(m)+n-m; n) ];
m<n
n) ) n+
= min(n:
~
a(n)
0) ;
1(n) ~ f(n-l) <=> En non-empty <=> 1(n) < aO(n) ~ 1(n) + 1
(13)
iii)
(
fen)
= t(n-l)+l. <=>En
(aO(n) )€[a 0// ( a(n) }]);
empty <=.> t(n) < aO(n) ,
iv) E non-empty -=> fl\(n)-l. < t(n-l) < aO(n-l) •
n
-
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CHAPmR II (Linear Absorbing Barrier)
II. 1 - In this Chapter we shaD. examine pr:incipa1ly the general scheme
R.~'l.
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is linearly variable (especially far 0 < s < 1):
~',
x -x
(A(n)} == (Sn+A) <==> (a(n)} == (sn+h) wi tIl s =
11 =
A
xIf -x ,>
-8
(h
<
-13
,
> n+=1).
It llou1d be rrx>re exact to spew;: of "a. b. variable according to t~le
ariti.l1JCtic progression (a(n)} = (sn+11) " because we are interested in the
values of the linear function
,,'e
sllaJ.l !:eep the notation of
st+h for t=l; 2, ••• •
linear~r
"a.. b.
restriction just now indicated t=n=1,2, •••
Then in the case where a(n) = 8n+h
aI
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[(x':x");x(n) < A(n)] == R.:-r.[(O.l);f(n) < a(n)] i'T:len the upper a.b.
Howeyer in the foD-ouine:
variable
aCt) ==
•
(h > -s) (13) becomes
i)
0 ~ fen) ~f(n) ~ n~ [fen) : fen) < ns+h] ;
ii)
11 < n < -1
h + 1,
no = min ( n:n -> -11)
1 <==> -1
-8
-8 - 0
-8
8 < 0 <=> n+ = min(n : n >- -
(a(n)}tl ,nth tIle
!!)
S
<==> - !!
< n+ < s-
!!S
+ 1-/
a) 8 ~ 0 and (14.i) ==> 1(n) ~ f(n-1)<=> En nOn-cll1Irty,no
(14)
b) 0 < 8
< 1 and (14,i)-=>
:s n ~ n+;
either 1) 1(n)=f(n-1)<,-=> E nal-emp"ty I
_
_
n
.
{ or
2) f(n)=f(n-1)+1<==> E empty;
n
c) 1 < sand
-
{ 0 < s +11 < 1 --> 1(n)=n-1 <=> E
-
iv) E non-empty
n
> fA(n)_l <f(n-1) < s(n-1)+h
-
> sn+h
Since for any integer
1'-1 < s(n-1)+h, and
such that
> 1(n)=n <==> En empty far every n ~ 1
s +11 > 1
n
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empty far every n
> 1;
~
~ f"(n) < sn+h+1 - s.
:r 8uch that sn+h <
l'
< sn+11+1-8
(s
< 1) we
have
since from (14,iv) there exists al~rs f(n-1) (~f(n-1»
r A {n)-1=f(n-1) ( < s(n-1)+h) ''1e obtain
11
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(15)
sn+h
values
~
integer f < Bn+h+1-s
~
rJ'(n)=f
(-> f'\n) takes all the integer
belonging to the iIItervaJ. I(s). [sn+b, sn+h+l-s[
f
Espec1~
I
with s < 1 ).
we have the following &SBerticn:
" If (and only if) 0
~ 8 < 1(<=> r"(n)
f~n),
! 1 JiI(S») then
when is
defined, take only an iIIteger".
Let us consider
Remarking that
8 -
d-l
)
with d=1nteger > 0 (
0 ~ a· T < 1 •
dd1 nil
d~l nl+c-
<-> n"-nt=td <=>
d-1
T
c-t(d-1)
if (and only if) c-
¥
then if f"(n) is defined for
(nfl_n l ) <->
n=m we have
d 1
) +h ~ f ~ (m) +t(d-1) integer < d
d-1 ( m+td ) +h+1-s for t=O,! 1,
T<m+td
+ 2, ••• and m+td
-
Since by definition [(6),(14,ii)] there exists
n -m
min t=
T'
tA(n)=n =-=>
n -m
it we put
and to
nw-Il
T. r
> n0 •
-
0
0
we ha\e
dd-1 (n +rd)+h < n + r(d-l) < dd
..1 (n +rd)+h+1-s ~ t"(n +rd).n +r(d-l).
o
- 0
0
0
0
So we obtain
(16)
s= d;l (d integer> 0) <;:=> I'(n )- n -r , tor nr*llo+rd and r-o,1,2, ...
r
r
where n -min (integer m : m > dh) ,
o
-
and this 1Dij;>lies that the linear a. b. ~;1 n+h} is equivalent 1:0 the a. b.
II.2 - Now let us investigate the problem of the absorption probability
(17)
P
If
= pr (3
n:
"t(n)"
2: sn+h } •
s < 0 from (14,ii) we have
h
h
-:s
~ n+ <:s+ 1:
If s=O the distribution of the r. v. lin"
and the distribution ot the r. v. "n-no"
binomial distribution : as known
12
is
(no=m1n(integer
hence p. 1.
Pascal's distribution
2: h» is the negative
J
I
I
I
I
I
I
I
et
I,
I
I
I
I
I
.-I
I
I
~
I
I
I
I
I
I
I
lit
I
I
I
I
I
I
I.
I
I
=p
P( n )
If' s ~ 1, with
because
sn+h
> nx"
n
0(n-1 )
n -1 q
o
a(l)=s+11
= n.
> 1,
n-n
0
P-1
,
the barrier (sn+h) is not absorbing
Therefore
o.
P =
> 0), we have n+=l, that is
If s ~ 1, with s+h ~ 1 (obviously s+h
have absorption only when f'\(1)=1
For the cases in which 0
sche~
to WaJ.d t S
P~l=P.
and therefore
< s < 1, h
we
0, our r.w. scheme is equivalent
~
of a SequentiaJ. Plan for a Binomial Distribution with an in-
decision zone of the frequencies f(n) on the Cartesian plane [s, f] included
between the two parallel straight lines
< 0 < h, when ho
the following we put P=I>o if P < s, P=Pl if P > s.
L e (f=sn+h) and L e (f=sn+h) with h
0 0 0
In
NO\'T
vTe
n-sn-h
ql
n-sn-h
sn"h n-sn-h0
0
ql
1
n-sn-li
sn+h
0
0
Po
%
p
=A ,
~
0
_=>(~)S (~)l-S
~
=>
=B
obtain the follo"Ting relations
0 < P < s < P < 1, h < 0 < h :{A
1
o
B
(18)
co
let us make the folloi'Ting ratios constant
sn+h
Pl
sn+h
Po
from 'uhich
-+ -
~
r
for each nt>
(~rS ] (~~~)O for each ~J
1
= 1
==>
P1%
~~
0< B= (P1~~~ )ho = (~\f
~r
The equation
((~ (~rr (:~~
= )
= ~~)
=(~)s
~
>1
<1 <A =
__>
l ~)h
(P~~
=
(~)s (ll-X)l-S=l for 0 < s < 1 gives always two real
P
-p
and different solution (coincid:i.ng if p=s ) Po
13
and Pl
wi. til Po
< s < Pl ;
I
hence if p
or P=P1 > s,
given as either P=:P < s/We are interested in tlE other
is
o
solution g(p,s).
But
there is not an algebraic expression giving directly
P1 if P = P
g(p,s) = { P if P • po
o
as a function of p
and s.
However in practice,
1
to the solution
in eveIJT concrete case, we can get numerical approximation
g(p, s).
Let us then assume that
immediate1~T the
some finite
is PE"&ctically known; we will get
g(p, s)
values of ~(h) and :&=B(h ).
o
n would be for the first time
The probability that for
f(n)=l'(n) ~ sn+h
(from
(14, iv)
f"(n) < sn+(h+1-S», with sm+h < f(m) < sm+h for each m < n, :from [1] is
o
1-~ (h )
l-~(h )
(19)
a
i'(h+1-~)' < a = a(ho ) ~ A(h,o
when p = Po
{
(3~(h ) < B(h )
o 0
1-(3
The straight line L
o
.E
p = P1
when
sn+h (h < 0) is a lower bc:und put in the
0
0
initial R.W. [(O,l),f(n) < sn+h] which can be considered as limit of the
scheme
R.W. [(0,1),
Then, because
==~
(3 (h0 ) =
-11 lim
~ _00
o
sn+h < f(n) < sn+h] when h ~ - 00
o
0
•
~1~)
P1Clc > 1 and :from this h lim B(h )=1 lim
h0=o=>
Poq1
0 ~ -00
0
lO~ -~oq1
°,
we have
P • h lim
-+ .. 00 (l-(3(ho
o
»
that is
(21)
{
p q
(...2..1) h+l-s < P <
P1%
-
PoQ1 h
(p-q)"
ifp=p <s
0
P = 1
If Po =:P = s
1
then
if P
PoQ1
P1~
= P1
=1 and we have
Recapitulating all cases examined about the
14
,
> s
P = 1.
co~le
(s,p) with
J
I
I
I
I
I
I
I
til
I
I
I
I
I
I
.'I
I
I
'-
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
<
- 00
S
< +
0
00 ,
~
p
1, we can write
~
P(s):l
P q ,
P = p(s)=p(s;p):
(22)
far p > s
P q
(~~)11+1-S< p(s) ~ (p~~)h for P=I'o < s < P1 < 1,11 2:
p(s)= 0
d-1 n + d
no
f ""'(n) = d
p
=
--:;:.1_
n
A
Hence
(23)
(t>
2: 1 •
far s
d-1 ( d = integer> 0 ) from ( 16)
s =d
I f ue consider
(s < 1) ,
f ar n=nr=n + r d ( r=O,1,2,....
)
o
't'Te
have exactly
·
Then ( 2
0 )g:J.ves
exact ly
•
"Ie have
(
d-l
d-l
Po~
P=I'o < s = 7=>P(s) = p (7) = Pl%
For instance, i f d=2 we have
and therefore
P=
~no
R.W. (O,l);f(n)
•
s
1
= -2=> p o
no
T.
1)
Poql
n 2.
= 1-Pl==> ( - p)= (..) l P=I' <-2
1%
q,
0
We remark that
<~n+~ J=
)
.
R.W.[(-l,l),z(n) <no]
is the scheme usually used for the classical problem of the gambler' s ruin who
l1avine
n
o
is playing, 88ainst an infinitely rich adversar"J, a series of
games in each of which P is the constant probability of losing the stake 1.
II.3 - In all cases in which P=l (P=I'l
average ii='ii(s,p)
of the number n
2:
s)
we have the problem of tIE
of steps required for a.bsorption.
can again utilize the scheme of WaJ.d· s Sequential Analysis
emp1o~Ting
We
the
Average Sample Number function of the Sequential Probability Ra.tio Test.
'1
After putting P% = (~)S = eU Where from (18)
poq
q
1
(~) s > 1. > u
PCP1 > P ==>
o
ql
equivaJ.ence
R.W.
~O,l);f(n)
< sn+h]
=
l!
~ 19 ~
>0
q1
(s > 0), let us c alS ider the
R.W. [ (z',z";z(n) < hu
15
J.
0 ,
I
J
From (9,i), where A(n) = sn+h, C(n) = hu, we have
==> usi'V'=O => V=
u(sn+h)+vn=hu
z'
i)
zn
=
(24)
i
= ux'+v
t
= 19
I from
l'lil1ch (x'=O, x"=l)
= -us
ilith proba.bUity q ,
'"0
P,\>
z" = ux"+v = J.c + 19 .S1.. = 19 R... = u(l-s) '\'rith proba.bi1ity p.
~
poq
n
i1) zen) = 1: zi
i=l
Therefore
(25)
~
= I.e
-su
= f(n)z"
Po
'
+ (n-f(n» z' < hu •
'{(lg t-, 19
R.l'1. [(0,1); fen) < sn+h}ER.
~
R...); zen) < 11 18 -~] •
,
Po
s
q
Besides we have from (8,ii) and (24,i)
zA(n) = l"'(n)(z"-z') + nz'
> 1m =-> ~::r
fA(n) u
ns ,
and af'ter from (14, iV)
11
< z"(nl = :r"(n) -ns < 11+1-s ==>
-
u
> hu
~ E(nz
hu
< zl'(n) < (h+1-s)u =>
-
(n)") < (h+l-s)u •
Remembering the Wald' s equation [1]
we obtain the following interva.l of estimation of the average number
= ii(s,p) :
s < p:
(27)
where E("z,,)
= ZIp
~ < n(s p) < (h+l-s u
E\"z"J - '
E "z"
+ z"q
= 19(T)P
o
But if
s=p
we obtain
=hlim
( .;-)q , u=lg
s=p.
~)=2
Po"'P E\"z"
0 ,
1\
:~
•
0
'"0
Ii = co... Tn fact
I
I
I
I
I
I
ea
J\
n = E( "n,,)
I
from which we have, following 1 'Hopital's rule,
-> p o
= P =>slim
... p
E1 "z" )=
I
I
I
I
I
I
.-I
16
I
I
~
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
n(p,p) = h
=
dp
lim du
po"'· dpo
-pqh
lim
E... O
0
dE("z,,)
1
=
E
= h
(
0()
lim
po... p
-pq1
1
l-p _ 1?..
l-p
Po
0
=
E....
> 0)
pq
that is
(28)
n(p,p) = n(s,s) = •
F or
d-l (d tnteger
s =d
> 1)
d-l and' no = nu.n
. (int eger
>d
when p
~
dh )
we have exactly
n
(d~l,p) =
For s=O
no
E(flz,,)d 19
p~
poq
•
(d=l : Pascal. 's distribution) we have exactly
n(O,p) =
Ho't'lever for
•
s < 0, except the obvious and useless inequality
.....h...
< Ii (s, p) < - ~s (lh-s and - E
are the the abscissae of the points of
l-s
s
contact of the straight line
f=sn+h
with the straight lines f=n and f=O)
there are no synthetic methods able to eive a good estimation of n (s, p).
In the following (III-2.)
we will see that it is possible to use in this case
(s < 0) a direct analytic method which gives the exact distribution of
n
from widell we obtain the exact value of n(s,p).
II-4.
If now we consider a schenJ:! R. W. [(0,1) ;f(n) < a(n)] ,-there
(a(n)}
is an whatever a.. b. not linear ftmction of n, in sane cases the results
obtained above may be used witIl a.dvantage according to the validity of the
approximation which will be
n~r
discussed.
17
I
J
'ive remark that the probability P of absorption by the a. b. (a(n)} as
ii (if
well as the average number
P=l) is a function of the sequence
».
: P--P ( (a(n)})" ii = ii «(a(n)
(a(n»
1'-breover from (10) and (13) we have
(31)
P=P({a(n»)
II:
P«(aO(n»)
for each a. b. (a·(n)) e [aO/ / (a(n)
P=l... ii( (a(n»)) = ii( (aO(n) »
Given two a.b. 's
of the events E and
P"=P(E")=P( (a"(n»)"
(a'(n)} and (a"(n)}
i
»).
"remenbering the definition
(I.l) and putting P'.P(E')=P«(a'(n)})"
P(f')a:l-P', p(E")=l-p",
we
have P' -P"=P(E")-P(E') •
If there exist (a·'(n)}e(a·'//(a'(n)}] and (aO"(n)}Et(a·"//(a"(n»
such that
aO'(n) < a·"(n) for each n (> n'-min(n:aO'(n) < n)
..
-
0
-
]
then for
every sequence {f )... (f(n)} such that fen) < a 0 , (n) far each n, we have
n
also fen) < a O"(n) for each n; but the converse generally is nat true : hence
aO'(n) < aO"(n) <;n:> «(f }Et
-
n
P'_P"=p(E'''_i t )
THEOREM 1.
2: O.
COROLI..ARY 1.
n
~> E' c E"
-
"
from which
This result is summarized in the following
If there exist two a.b. 's (aO'(n»
to the given a.b. 's (at(n»
for each n
i'=> (f }Et E")
and
(a·"(n)} equivalent
and (a"(n)} respectively, such that
aOf(n) ~ aO"(n)
(> n'),tben p'=p( eaten»~) > P"=p( (a"(n»).
-
0
-
i) (for each n"
3 mn > n
such that a·"(m ) < aOt(n)]
n
-
=>
_-> i'E E" ==> pt=p"
ii) P" = 1
==> pt = 1.
So, given the a. b. ( a(n)} , if it is possible to find an equivalent
a.b. (aO(n»
s'n+h'
and two straight lines a'(n)=s'n+h'
< a·(n) < s"n+h"
-..
(32) pt
2: P 2: P"
, where
for each n
> nt,
0
..
and a"(n)=s"n+h" such that
then we have
p=p( (a(n) }), pt=p( (s 'n+h'}), P"=p( (s"n+h"}),
with P' and P" available from the above considered methods (II. 2, (22».
18
I
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I
I
I
I
I
--I
I
I
I
I
I
.-I
I
I
\.
I
I
I
I
I
I
I
lit
I
I
I
I
I
I
Hence CorolJ.a.r'lJ f. ii) and (22) Give us the following
Given the a.b. (a(n», if there exists an equivalent a.b.
THEOREH 2.
(aO(n»
E E Ea
PuttiIl[;
(b(n»
(a(n», let
: a' (n) _< a(n), E 'E E ]
a
a
Then from CoroJ.J.a.r"lJ 1. i) l1e have
is nondecres.sing.
Ea --> P=P«a(n) ))=p( (b(a) »=p( (bO(n»)
for each (b°(n»
[bOil (b(n»
€
]
be the lo,ver extremum of the class [b °1/ (ben) )]
let {b(n»
HOlT
no)' and
= min [ (a' (n»
r:1
i) R.W. [(o,l);f(n) <bO(n)] ~=> 0 ~ fen) ~f(n) = b(n) ;
ii) (b(n»
= L{a,(n»,
where
L is the law of the correspondence (a(n) }-.O:>(n»)
(ben»~ does not belong to the class [bOII{b(n»)].
From (10) we know
In this 'Hay the greatest inferior bound for P is
max (1)" : p" < P] =
-
-
~r
definition ( b(n»
max (1)( (s"n+h tr »
S" h rt
,is
: s"n+h" > b(n)] < P.
-
nondecreasing sequence, "Ti.th
b(n+h)-b'(n)=O,l and{ n~(n)) nondecreasing : hence the upper limit of the
{btn ) } is always defined, and we have
sequence
n
0<8
rrm
=
n .... ca
<1
b(n)
n
-
•
Moreover by definition of (aO(n)}, (bO(n)) and (b(n)} we have
bO(n)
n
> -s
rrm
'
a O(n)
ii -
2:
s
for each (b°(n)}
€
[bOil (b(n)}
] and
n .... ca
( aO(n»
E
[a
II{a(n» ].
Therefore
s = min [sit: s"n+11" > b(n),
Having found
I.
I
I
when tile a. b. is
b(n)=mll1 [a(m):li1 > n] <--"> ( b(n»
the seqt'l.ence
2:
sit --> p"=p «(sltn+h lt ))=l => P=l •
2:
if p > S", then we have: p
~5
s"n+h" 2= a (n) (n
a.nd. a straight line (s"n+11") such that
{b°(n»
€
h~
[bOil (b(n)}]
h" <
ca].
such tl18.t sn+h~ > b(n) there exists aJ.uays
such that sn+h" > bO(n) > b(n) for each n(> n ) •
0-
-
Since p( (b°(n»)) = p( (b(n») = p( (a(n»)) = P we have from (32)
19
0
I
P
J
2: P" = p( (sn+h"». So Theorem 2 becomes the fol.low1ng
THE<m:M 3.
=L{a(n)}
(ben)}
, then P=P( (a(n)
Given two a.b. 's (a'(n»
P"=p( ( a"(n») =
==
and (a"(n»
~
~~~:~}
P"(n), with
n-l
{ (a'( )}
steps by the a. b. (a"(~».
after n
n
n ....
»= 1.
=
~
2: s =- iIiii
Given the a. b. (a(n)}, if p
where
'
==
GO
, let P'.p( (a'(n»)
= ~ P'(n),
abso~~~n
• probability of
.
Under the conditions of Theorem 1 the conditional. probability P"(nlm)
is defined ,that for some sequence (f(i)} there occurs
when we Im'Ve
f(m)=t"", (m)
'"
P"(n 1m)
=0
> a(n».
for m > n , m < n~ and n < n~.
n
i) P"(n) =
L
2: aO "(n)
2: aO , (m) (we remeber that in the same sequence
(f(i)} , f (n) is the first value for which fen)
Obviously
f(n).t'''(n)
L
So we have
GO
P"(n I m)P' (m)
,
11)
m=n.'o
P"(n I m)=l ,
n=m
•
L
and we can consider the conditional eJq?ected value ii"(m) =
nP"(n 1m)
n-m
Hence if P'=p II=l, putting ii'=
•
L
•
-:. L,
n" =
•
n"(m)P' (m)
o
n( (a' (n»)
•
L L
•
~ L
nP"(n)=
n
n=n"o
n==n"0
m=n
n
and
n".n( (a"(n»)),
L
P"(nlm)P'(m) =
P'(m)
m=n'o
m=n'0
we have
L
GO
nP"(nlm) =
n=m
•
L
mP'(m) =
nP'(n) = ii'
m=n'o
n==n'0
,
from which we obtain the following
THEOREM
4.
(a 0 , (n»
I f there exist two a. b. 's
to the given a.b. 's (a'(n)} and (a"(n»
aO'(n) ~ aO"(n)
for each n(~ n6)
and (a 0" (n»
equivalent
respectively such that
and if P'=P"=l, then ii' < ii" •
20
~ me
I
I
I
I
I
I
I
til
I
I
I
I
I
I
.-I
I
I
~
I
I
I
I
I
I
I
and (32) we have
p=p'=p"= 1.
s"
(2: s I)
from (22)
Hence, remembering (27), we obtain the follow-
ing
THEOREM 5.
a.b.
Given the a. b.
(a(n)), if it is possible to find an equivalent
(aO(n)) e[ao//(a(n)} ], and t"TO straight line
such that
i)
ii)
(s'n+h')
s 'n+h' < aO (n) < s"n+h" for each n > n ,
-
p
2:
sI
(2:
-
sIt
-
-
2: ~n
ae
I
I
I
I
I
I
I.
I
I
2:
Under the conditions which e;ive us (32), if p
21
we have
0
and {s"n+h"}
I
CHAPl'ER
m
(ITon-linear Absorbing Barrier)
Let us observe that when in R.W. [(O,l);f(n) < a(n)] the a.b. (a(n)}
III.!.
is nat linear, its trend may be too far from a linear one:
(32) and (33) we shall have values
and
n" - ii' = (11"+1-8"-h')
Sf,S",
in this case in
he, h" implying differences
pf_p" > 0
E(~Z,,) > 0 which can be too wide for a more
precise estim&tion of P, if P < 1, and of ii , if P
= 1.
The following cases are the ones in which the same problem occurs even
when
the a. b. is linear (sn+h):
i) s < 0 and we ask for ii (P=l)
ii) s
> 0, but a truncation of the r.w. takes place when after !! steps it
has not yet reached the a. b. (ns+h) (the two events E(!!) "absorption before
the first ,a steps" and E(,a) "truncation at the ,a.th step" are necessary and
incompatible).
Such an a. b. is not ]j.near: it coincides with the arithmetical progression
(sn+h)
Due
n
t
i=l
P(i)
o~
for the first n terms.
to the truncation, the probability of absorption will be
= p/... ,< P;
Et
where P is the Imown probability of' absorption when no
truncation occurs (!!. = 01), and the difference
so that, if ,a is not
sufficient~
p-p/-gJ increases as n decreases,
large, an evaluation of
p/-gJ by P
may
lead to coarse errors.
However that may be, in order to analyse non-linear schemes it is mre
convenient (or necessary, depending on the situation) to
app~
other analyti-
cal methods 'which give the probabilities Pen).
T"nen from the sequence (p(n)} we can further calculate
direct~
P (if P < 1) or the average number Ii and other characteristic features
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Moreover to settle dire ctly ''1hether P=l we can use Tlleor ~m 3 fram
(if P=l).
Chapter II.
Genera.lly, following (1.3,iv), the event
of ken)
En
'Hays according to the values
is emPty
n
can be realized in one
f~(n)=f~(n)+i-l (i=1,2, ... ,ken) (if
put k(n)=o and there is no value l'(n)= f(n-l)+l).
''Ie
Let E
E
.
be the single event corresponding to
n,~
A
f.(n)
~
and C(n;i)
the nunber of arrangements of n displacements in which there are
placements equa.l to
A
f (n) disi
f" (=1) "I'r.i.thout obtaining any fA(m) fur each m < n:
then ,·re bave
n
~(n) n-~(n)
c(n;i) ~ (~(n)}; P(n;i)=pr(En;i) = C(n;i)p
q
•
ken)
~
P(n;i) (the ken) events
i=l
En;i are incompatible) can be reduced to a practical n:ethod for the calcula-
Therefore the calculation of Pen) =
tion of the coefficients C(n;i).
nI-2. - The one-dimensional R.W. [(0,1); fen) < a(n)] corresponds to a
particular case of two-dimensional r.w. in the semi-infinite first quadrant
of the Cartesian plane [t,f](t ~ O,f ~ 0) bounded by an absorbing line (a. b.)
whose function is
f=a(t)
~
0 for
a(n) = lim aCt)
t:::;n
t
> 0 such that
for each n=1,2,...
•
The initial position of the two-dimensicnal r.w. is at the origin
~(O,O),
and its system ot displacements at each n.~ step is represented by
the rand.an vector tit" = "f" = (1, "f,,) which take the vector value
-n
f = ft = (l,f t ) = (1,0) with probability q=l-p and the vector value
-n
!n
= !," = (l,t")
= (1,1)
with probability p: that is pr("!n" = (l,f» = pfql-f.
n
4. It follows that the two-dimensional r.w. can
i=l
take on1¥ the posi tias defined by the vectors or points !( t) = (t, f( t) ),
Let f(n) =
~
23
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with integer coordinates: t=n=l,2, •••• ; f(n)=O,1,2, ••• , fen) (see (13,i).
At each step one can go only fran a point or vector !(n)=(n, f(n»
to the point !(n+l)-(n+l,f(n+1J, where
-+f(n+1)=!(n)+!n •
n
Therefore each position fen) can be reached if and only if the number
of steps is n.
ing the sequence
f(n+1)=f(n)+f
M:>reover a necessary and sufficient condition for consider(f(i)} -+ f!(i)=(i, f(i»}
a sequence (Oi) of n
steps or trials
f(i+1)-f(i)=f=O,1 for every i=1,2, .. ,n:
(i-1,2, .. ,n) as the result of
0i (Oi -+ f
lie
-+ f(i» is that
i
shall call such a sequence
tlsequentialtl with regard to the system of displacements (ft, ftl)s (0,1).
Any particular arrangement of the two displacements ft -+!'
and
ftl -+ ftl
which are repeated at random in the same sequence (f ).. (f(n»,
n
can be graphicaJ.ly represented by drawing segments beween successive points
of the sequential sequence ( !(n»
==
((n, f(n») •
We obtain then a nCll-decreasing graph with horizontal unit segments
(parallel to the n-a.xis) and
u;plmrd
unit segments (parallel to the straight
line f=n) which correspond to the vectors ~ = (1,0),
.!; =
(1',1) respectively.
Such a graph we wUl call a "sequential step graph.. (s.s.g.) starting
from the origin .Q. : it will be said of order n if limited to
n
consecutive
un!t segments.
Given a point (n,a) (0 ~ a = integer ~ n), the s.s.g. t s of order n
which end at it (1: fi=a)
will be called ..converging to (n,a)", and the one
which originates from (n, a) considered from the (n+1), th step,
..starting from (n,a)...
The number of s.s.g. 's of crder
n
~11
be called
converging to
(n,a) (starting from.Q) is the binomial coefficient (:), i.e. the number of
arrangements with repetition of n
elements of which a
are equal to fn
(n-a) are equal to f'. The probability of each of these (:) s.s.g. 's is
n
f l-f
n P i q i = p a qn-a •
i=l
24
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Obvicusly
o ~ a--f(n)
~
a < 0 or
(n)=o <r-==>" eitIler
a
a > n" gi Yes tl~ condition
n which is satisfied by any s.s.g.
Every s. s. g. which, starting from (n', a' ), converges to (n, a) is
equivalent to a s.s.g. of order n-n' : so the number of such s.s.g. 's wil1 be
n-n')
( a-a'.
Of course
0
~
a_a'
~
n-n'.
It foll.ows iDmediate1y that the number of s.s.g. 's of order n which
converee to (n,a) and go through (n',a') (i.e. converging to and starting
is the product ( n',)
a
Let«~,ar») be a system of
from (n',a'»
( n-n',).
a-a
N points (r=1,2, ... ,N) such that for
every point (nr,ar ) there exists at 1east one s.s.g. £.!:(n») which goes
through it (f(n )=a) without going through any other point (n.,a j ) of the
r
r
J
system lr.i.th n j < n
(n and a are integers). The point (n ,a ) is the
r
r
r
r r
first point of the system met
b~·
the s. s. g. £.!:(n»).
If 'l're limit such a s. s.g. just to the order n , we can say such a
r
s. s.g. is absorbed by the system in the first point (~,ar) to "fnich the s. s.g.
converges:
the system
«nr,ar )) as well as its points, will be called
"absorbing".
Then we can say that a point (n , a ) of the absorbing system is
r
r
"following" the point (nj,aj ), which
n -n.
r
> a -a.J > O.
J- r
is "preceding" the point (nr,ar ), if
We will call "subordinate" to the poin t (n., a .) eve ry
J
"follo1tring" point (n ,a).
r
r
J
For if the point (nj,a.) absorbs, all those
J
s.s.g. 's 'lMch go first through it, will stop at the order nj , and will not;
go on and converge to any absorbing point (n , ar ) following the preceding
r
point (nj,aj ).
If there are no other absorbing points (nj,aj ) preced.ing the point
n
(n ,a ), this non-subordinate point is said to be "primary" and ( r) s.s.g.
r r
ar
of order n converge to it. If then (n ,a ) 1s not prima.:r"lJ, the nwnber of
r
r
r
I
n
C < ( r) since some of the s.s.g.'s which
s.s.g's absorbed by this point is
r
ar
would have been absorbed by (nr,ar ), if it had been primary, do not reach it
since they are absorbed by the absorbing points which "precede" it.
n
r
Tllen, if (nj,aj ) precedes(nr,ar ), among the (a ) s.s.g's witich would
r
converge to the point (n ,a ) if it was prinary there are instead C j
r
r
~
ones absorbed (and stopped at the n
order) from the preceding point
j
(nj,aj ); C j is the number of s.s.g's of order n which converge to (n ,a )
r,
r
r r
going through (nj,aj ).
.
Since Cj is the number of s.s.g's absorbed by (nj,a j ),
,n n )
and(a::a~
the number of s.s.g's starting from (nj,a,,) and converging to (nr,a )
r
without restrictions, we have
C
n -n j
r,j
If
'\'Te
• Cj (ar_a
r
j
)
•
indicate by "h' h=l, 2, ••• , t r , the subscripts
j
of the t
r
points (nj,a j ) preceding (nr,ar ) and we put "t +l=r, we have the following
r
relation
t +1
r
L
t +1
t
h =l
from which we obtain the following recurrence formula for the coefficients
11=1
(34)
Since the C s. s. g' s 'Which are absorbed by (n , a ) are incompatible
r
an-a
rr
and each has probability p r q r r the probability of the absorption event
(nr,ar ) is
a n-a
Pr =Cr pr q r r
•
26
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B:/ detinitioo of "absorption" the N absorption events{nr ,ar ) of
the system
({n,a)} are incompatible, too.
r
r
of absorption by the system
({n,s.)} is
r
P =
If the
r
rr
I
L
N
(;6)
There fore the total probability
Pi =
C.J!Parqnr-s.r •
r=:J.
r=l
N absorption events (nr ,s.r ) are also necessary
'He
have P=l:
in this case we say the absorbing system is "closed".
For instance, if n =n constant for every r, then for the system to
r
be closed it is necessary that
a =r=O,1,2, ... ,n (N=n+l).
r
The points (n,r)
are primary (nQ1j subordinate to each other): therefore C =(~), and so
r
has a binomial distribution.
If
N < co (N= co) the absorbing system is finite (infinite).
"asyIli)totica~ closed"
r
We call
an absorbing system which is closed (P=l) and
infinite (N=co ), with a
> 0 for every finite r and
lim
a r > 0: the
r
r -+co
r. w. convi'rges to the absorbing system almost certainly {N=eo, P=l, but a
sequence
(f{n )}
r
impossible event :
III-;.
with f{n ) < a
p{i)
't'n1en a sequence
r
r
= 0 but
E
for every r finite or infinite is a non-
is not empty).
(a(n)} is given as
a. b., such an a. b. is equivalent
to the system of actual~ absorbing points «nt,at )} (t=1,2, .. ,N) located
upon or be'ji'ond the a. b. (a(n)}. We have at = integer ~ a(nt ), but the
same value
t).o -+ n
ti
n
can correspond to
k(n)
=n -+ a.t ~ a(n) ==>at = f~(n)
i
i
).
different subscripts
t:
, i=l, 2, ••• , k(n) •
So the absorbing syste~n (nt' at) ) this time is exact~ the system
«n ,f~n» where i=1,2, ... ,1.(n), n E (n) <::==>k(n) >0 (r=0,1,2, ... ),
r 1. r
r
r
n FJ. (nr ) <==> k(n) = 0 ~-> En emptJr (no f"{n) is defined for n) and
27
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f(n r ) < a(n)
:s t!'<nr ),
J
f~(nr) < 1';+1(nr ).
Since from (13,iii)
"t(nr )
h such that f(n)
r < fenr ) < fh""(nr )=>3{t(1), 1'(2), ... ,f(nr »
i) t(m) .<rem), t(m)-t(m-1)=O,l for each m:1,2, .. ,nr -1 (1'(0)=0)
and
such that
i1) fen -1) < ~h(n )-1 <fen -1),
r
r
r
1'(nr)-1'(nr -1)=1" ,
111) f(nr)l::f~(nr) > f(nr ),
then the assumption r(~) < ~ (nr ) < 1';+1(nr )
1':+1(nr ) =
ti<,\-)
a
~(nr)
+1
l
j
1<,\-) + 1
=> kenr )
implies
A
= f k ( n )(n)
r - fenr ).
r
Obviously, from (13, i v) ,
but fron (13,iii) (E
~-
1 is eIIU?ty for i=l,2, ••• ,n -n 1-1) we have
r
r-
fenr -l)=f(nr -2)+1=f(nr -3)+2=..... =t(nr-l)+nr -nr-1-1.
Therefore
fA
l:(n)
r
(n)
r
= fen
1)+ n -n 1
rr r- ,
and from (37)
(40)
kenr ) = nr -nr-1 -'r(n
\~ r )-f(nr-1)\~ •
=-=
If we want an a. b.
((a(n»
=
==
= =
=
such that keD) ~ 1 (<==> En non-empty) for
every n of the range (no,n+) of the r.v.
"n", we obtain from (40) in
accordance with (13,i1i)
(41) nr -nr-1= 1 for each r=O,l,2, ••• ~> k(n)=f(n-l)-1(n)+1 for each n~>
<-> "1'(n-l)-l'(n) = k(n)-l ~ 0 <==> r(n) is a non-increasing sequence.
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(Conversely, if (f(n»
is a non-increasing secpence then from (1.3, iii)
we have E non-empty for every n such that n < n < n+).
n
0-
Let us consider then an a. b. (a(n»
a non-increasing sequence (f(n)} for no
such that to it there corresponds
.:s n .:s n+
~(n') = n' for n' < n).
This occurs, for instance, when the a. b. is linear
From (41), (:58) and (:57)
(42)
~re
-
(sn+h) with s <
have
f:(n)(n) = f(n-l)+l = f;(n-l) •
~
Hence every absorbing point
(n, fk(n) (n»
s.s.g.'s which converge to it pass through
is not pri.Ina.rJ and all the
(n-l,f~(n)(n)-l)= (n-l,f(n-l»,
which is a primary non-absorbinr; point because by hypothesis
r(m) > f(n-l) for each m < n-l.
(43)
C(n,k(n».
On
(f(::~») ·
the contrary, since from (37), (41) and (42) it follows that
A
A
fl(m) > fi(n)
(44)
Therefore
for each m < n and
every absorbing point (n, f~(n» with
r·breover from (42) if k(m)
we have
f;(m) = f{(n)_l (n)+l.
i
< ken), 0 < f < n -1
-
-
0
> 1,
i < ken)
i < ken) is prinary.
k(m+l)=k(m+2)=... =k(n-l)=l, ken) > 1
So
f."(n)=f ==> n=n(f), Which is a single-valued nonJ.
increasing function of f.
Therefore
(46)
i < ken)
: C(n,i) =
(f~n» = (n~f»
i
with 0
~f ~
no-l.
There are no other absorbing points besides the ones considered in the
two systems {(n,f;(n)(n»} (tlus set includes all the points
I.
I
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o.
29
~dth ken)
= 1)
I
and {(n(f),f)}
with no:S n:S n+, O:S f < no-l.
Hence N=n+ + 1.
The
union of the t"JO systems gives us the absorbing system of the r.w. scheme
cons idered.
Concluding, we have the following probabilities for the two types of
absorbing points
Because the sequence
(f(n)) is non-increasing the absorbing system
is al'WaYs closed (asymptotically closed if n+=N:oo and lim fi(n) > 0)
n=oo
Hence
n -1
n0 -1
n+
n+
0
(48)
p=
L
n=n0
Pn+ I
P =1
f
f=O
f=11o
n(f)P
f=O
constant, it is
necessari~
0
In the case in which
I
, n=I nPn+
n=n
f
•
true that
f (n)=no -1, k(n)=l for every n>n. Therefore
-
n-l \
C(n,l~(n»= C = (
) ,
n
n-1
o
and the distribution of
lin-no"
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0
no (' n-l )
P = P
1
n
no -
n-no
q
is the negative binomial distribution.
Since it is known that Pel, the infinite system of absorbing points
(n, no) } is asymptotically closed.
rn-4.
J
In practical applications in which the
scheme
R.W. [(O,l);f'(n) < a(n) ] becomes more appropriate, the case of a nondecreasing a. b. (a(n)} generally occurs.
In this case
we will give to the
recurrent expression of the absorption coefficients a farm ldlich is nore
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convenient both for practicaJ. reasons of swifter caJ.culation and for theoreticaJ. reasons of facilitating a mare detailed analysis of such coefficients.
To a nondecreasing a. b. (a(n)} ther:e corresponds a nondecreasing
sequence (f(n)}.
So in the follo,·r.i.ng work we shall consider the equiva-
l
lent a. b. (aA(n)} E a all (a(n)}
Since
(n, f(n-1»
J
where
is the upper
aA(n)=f(n) + 1.
s. s. g. which is not absorbed, we
call the equivaJ.ent a. b. (a"(n)} ~ (f(n)+1)
From
(50)
"sequential".
(37) and (38) we obtain
ken) = f(n-l)-f(n)+1,
and since (f(n)} is a s.s.g., from (1.3,iii) we have
_
( 0 ~
_
=~
f(n)-f(n-l)
-> k(n)=1 <==> n E{nr ) <.=-> f(n-l)+1=aA (n)=f"(n),
l1 <==--> l~(n)=O <-: > C(n)=O (f"(n) is not defined).
So the nonincreasing absorbing system {(nr,ar )} (r=O,1,2, ••• ) is
defined, l'lhere
a =a."(n )=fA(n )=f(n -1)+1=f(n ) + l ,
r
r
r
r
r
a =n
00
•
Then, since n 1< n
and a 1 < a , (:;4) gives us the following
rr
r- - r
recurrence formu1.a for the coefficients C(n)
r
r-1
nr
\
n-n
(53)
C(nr) = Cr = (a ) - ~
ci (r i) •
r
i=O
ar-ai
In order for the absorbinG system (nr,a » to be infinite, the
r
sequence (n -a ) , which is positive non-decreasing, must be divergent.
r
r
finite non-increasing absorbing system
101ld
not have practical interest,
unless the event "truncation" is considered.
I f we put
k(n -i)=O <
r
from
n -n l=d > 1 then (51) and (52) give us
r
r-
r-
>f(n -i)=f(n ) -1
r
r
for i=1,2, ••• , d
r
lT~lich
31
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(54)
i) a r ~(nr -i)+i
(i=l,2, •• ,d),
ii) (i=dr and (52)~->ar =ar-1-1+d.
r
r
Hence :from d r =nr -nr-1=8.r -ar-1+1 we obtain by recurrence
nr -ar = nr-1-ar-1+1 nr-2-a r- 2+2 = •••• = n0 -a0 + r
=
and since a =n
o
(55)
we find the usef'ul relat10n
0
=r
nr -ar
for each r=o,l,2, •••
•
We note that
i)
(56)
{
ii)
n
r
= min
(n: n~A(n)
= r)
a r = min (a'n): n-t(n)=r ).
Then it follows that the l:ncn-rledge of the increa.sing seq)lence of
integers
(~)
is sufficient to represent a "sequential" a.b. (aA(n».
Letting c(nr ) = Cr '
"td.th the ne"t'1 asswuption (53) becomes
r-1 C ( nr -ni
) _ ( n r_
) r-1
L i n -r-n.+i - r I
r
i=O
and the probability P(~) = Pr
(58)
Pr
= pre
"n,,=nr )
Letting D(n,a)
= Cr-
1.
L
i=O
of absorption by (nr,nr-r) becanes
n r
rq •
'D
indicate the number of s.s.g. 's( f(l), ... ,f(n)=a )
which converee to the non-absorbing point (n,a) with a < a:"(n)~(n)+l and
f(m)
(59)
;s r(t'l)
for each m < n-1, v~ have
D(n,a) = D(n-l,a) + D(n-l,a.-l)
because if fen) = a. then necessarily f(n-1) = a-1, a.
We put C(n,a)
= D(n-1,a-1).
Since every s.s.g.
(f(l), ... , f(n -i), ... , fA(n ) = a ) "tm.1ch reaches the absorbing point (nr,a )
r
r
r
r
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necessarily passes through the points (n -1, f(n -i» =>f(n -i)=1(n -i)
r
r
r
r
with i=l, s, ... ,d , we have
r
C(nr,ar )=c(nr -i+1,ar -i+1) = D(nr -l,ar -1)
So by (54,ii) and (59) ire obtain
i=dr ~C(n,a
r r ) = D(nr-1,ar-1-1)
for i=1,2, ••• ,dr •
= D(nr-l-l,ar-1-1)
+
+ D(nr-l-l,ar- 1-2 ) = C(nr-1,ar-1) + C(nr-1,ar-1-1 ),
that is , remembering C(nr,ar ) = c(nr) = C
r1
(60)
Cr
= Cr-1
+ C(nr-1,ar-1-1 ).
This result will be use:f'ul later.
1II-5.
After choosing an integer r, let us make a first chanee in the
sequential a.b. (a<l\(n» by ma.1dng the followi~ change (dependent on r) in
the sequence (!Jl) -+ (aA(~)
= ~-h}
...
( n o,n ,··,n ,n s+l,n s+2,··,n r,n + ,··· )
r 1
1
s
where
i) s=s(r) such that n s < n0 +r < ,n's +1 -< ns +1
ii) n~+h=no+r+j for each
iii) a' s +j
j=l,2, ... ,r-s ;
= n~+j-(s+j)=no+r-s
constant for each j=1,2, ••• ,r-s
We put C~+j = c(n~+j' a~+j)
For the following developments
(61, i)
n's
= n0 +r
,
n'-s
s
'toe
also introduce
= a(n0 +r) = n0 +r-s
(j=O)
to "\'1hich there nill correspond the auxiliary coefficient
(61,ii)
C' = {C s if
s
0 if n
ns=n~ (n'= n +r)
s
< n'
s
S
0
•
C~
such that
I
,}
Applying (53) we have
n
C' = ( r) _
(62)
r
r
8
r-1
n' -n.
(r . H)
r-n
L
C
h
n
I
_
(
'-D.h
~-h
)
h=8+1
h=O
We now consider the difference
C'-C'
r
r-1
= t;)
_ ( n'r-1- n.'n)
r-1-h
r
_(n;_l)
_ ~ [(n;-~)
r-1
~
r-h
JC'h _( n'r 1-nr-1
' )
h=O
n;-~
Since, by definition,
_
n' -11 ]
( ~:i-h 1)
I (~-h )
r-2 [
011 -
n'-Db
h=8+1
0'
r-1·
= r-h
, n;_l-rth
= r-1-h,
n;_l
=n;-l,
on applying the known relation
(a ~ b)
(64)
we obtain
ir(~-h
n'-l-l1
(n;_l-m) =
i) ( ~-h) - r-1-h
nl-m
h.s s
for s < h < r-2, m;:::n.'
n
for
h= r-1, m=l'lh
1
ii)
n'
n'
( r) _ (
r-1)
r-l
r
--I
for~,
;)
n' -1
= (rr )·
So (63) becomes
s
n'
(65)
C'r - 0'r-1 = ( r-1) r
I
h=O
We then have for C; a. ne't'T eJq>res8ion as a. function of the first
values of 0h(h=o,l,2, ••• ,s)
,
s
1
r
0; = ( n _ ) (66)
r
h=O
L
(n;-l-~) 0
r-h
h
•
rle ncn'l calculate the finite differences of 0'r
1.st, 2.nd, ••• ,(r-s).th, applying (64) directly:
of order
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8+1
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n'-1
n'-1
A C'=C'-o' = (r ) _ ( r-1 )
r r r-1
r
r-1
s
n' -2
=(
rr
)_
L
l{n;-1-~)_{n;-1-1-~)J
r-h
h=O
C =
r-1-h
h
n'-2-~
\'
L
(rr_h 1)
h=O
s
n' -2-n.
n'-3
~ n'-3-~
(r-1
n)
= (r~l ) - ~ ( ~-l-h ) Ch
r-l-h
11=0
h=O
a
, 3
n '-3
A( 2 ) C' = A C' -A C' = C' -2C' +C
= (r ) _ \' (nr - -nIl)
r
r
r-l
r
r-l r-2
r
~
Ch '
h=O
r-h
n'
-2
A C' =e' -0' = (r-1 )r-1 r-l r-2
r-1
I
s
-I
A(2)C' =C'
- 2 C'
+C _
r-l r-l
r-2
r-,
n' -4-n.
11) C
r-1-h
11
( r
h=O
Ie
I
I
I
I
I
I
I.
I
I
A{r-s)
c' =
r
r s
-
I
.=
r-~
(_l)i{r-s) C'
i
n' -r+s-l
(r
) _
r
s
L
n' -r+s-l-n.
( r
n) C •
r-h
h
h=O
i=O
we have
{n'-r+s-1-n.
)-(r-h)=n -l-a(n )
r
n
0
0
:=
n'-r+s-1==> ( r r-h
~
) = 0
,
from llhich follows the relation
r-s
I
r=O,1,2, •••).
1=0
35
•
I
This gives us a new expression for
coefficients C;_l
(68)
=
c'
r
C; as a fu.ncticn of the (r-s) preceding
(far the definition of C~ see (61,ii»
(i=1,2, ••• ,r-s)
r-s
n +r-1
(o
)_
L
r
i=O
Putting i=r-s-j
-t j=O,l, ••• , r-s we also have
no +r-1
C'r =
Let us
00''1
( n -1
o
r-s-1
l:
) -
(_l)r-s- j (rjs)
.1=0
make a second change in the secpence
C~+j
(~)
•
so as to llLve
(70)
where
i)
ii)
iii)
11 is such that n~+j= n s +j for j=1,2, ••• , It
nIt s+j = n0 +r+j+1
a"s+j
Jf
=nIts+j -(s+j)=n0 +.r-s+1
S r-s-1
for jeI1+1, ••• ,r-s.
Ttle put C"s+j • C(n"s+j' a")
s+j and also introduce nIts+I = ns+I + 1 -tC"s+1 =0
1
1
1
Sin ce n" s+.1
n"s+j_1=n"
~~or each j =I 1+'
1 I 1+2, •• ,r-s
s +j1
= ns•+.1 +1 '
we obtain C~+.1 = c{n~+j,a~+j=n~+j-S-j)
= c{n;+1_1,n;+j_1-{S+j-1)1-1=a;+.1_1-1).
= c{n;+j-1,
n;+.1-S-j-1) =
So app~ing (60) to the sequence (70)
we have
where
") = c"s+j
.1
~("
~ ns+j,as +
,
e{ ns"+ _1' &s+j-1
" ) • e"s+.1-1·
.1
Therefore
(71)
e"s+j - cIts+j-1 • e's+.1·
36
,}
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Then applying (71) to the expression for C' in (69) we obtain
n +r-1
~1 ( l)r-s-j(r-s)C
C' - C'
- Cit
C"
( 0)
r - s+(r-s) - s+(r-s) - s+.(r-s-1) = n -1
j
s+j +
o
JII
_ \ (_l)r-s- j ( r-s)c
+
j
s+j
j=O
L
r-s-2
L
j=I +1
1
_ (_l)r-s- ( r-s-l ) (
and transferring
e"s+ (r-s-1 )
e"
C" r -
r-s-1
) Cit
s+(r-s-1)·
to the last member of (72)
n +r-1
(73)
r-s
Ctl +I = 0,
s 1
Remembering
s+(r-s)
=(
0
no -1
I
lie
obtain
1
r-s-l
r-s+1
(_l)r-s- j (r-s)c
_ \'
(_l)r-s- j (
)c lt
L
j
s+j L
j+1
s+j·
j=O
j=I +1
) ... \ '
1
Let us again make a chanGe (the third one) in the sequence (~) so as to
have
fI,
nil'
( no' n ,· •• , n +1 ' n , , '+1.+1
s+I +2' '" • • • ., n; r' .nr +1 ,·····
i
s
2
s
2'
)
2
"There
i)
ii)
I
2
for j=I1 +1,I1 +2, •••• ,I2 =s r-s-1
is such that n" .=n +.
s+J S J
nil'
s+j = n0 +r+j+2
}
for j=I +1,I +2, ••• ,r-s)
2
2
iii) a"~+J = n"~+j-(s+j)=no+r-s+2
. a'" .)
e'"s+j = C(n'"SofJ'
s+J
He put
nfl'
s+I
I.
I
I
I
r
=n
2
~r
s+I
+2 ... C'"
2
s+I2
and also introduce
= 0•
proceding in the same ,-my as in the previous case of I 1 , we find
37
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J
(74)
A) l~ make a sequence of changes in the sequence ( ~) so as to have
., .
(k+1)
(k+1)
(k+1)
. , ..
(no'~'''''' ns+~' n S+~+l' nS+~+2 , .... , nr
' nr+1'· .. ···)
in confo:rmity ,,11th each component of the non-decreasing sequence of integers
{Io,:s.,~,.... ,~, ... ,Id}
(far le=1,2,:3, we I1ave used tbe superfixes " ", ''')
where
(k)
i) lie is such that n s +j
= n S+j
for j=~-l+1,~.J.+2, ... , lie ~ r-s-1,
(k+1)
(k)
ii) ns+j = n s+ j + 1 == n'+j+k=
n0 +T+j+k
8
iii) a(k+l). n(k+l)_ (s+j) = n +T-8+k
s+j
s+j
0
with
1 0 = -1
, Id-r-s-l ,
(aA(no +r) == no+r-s) ;
d=e/'(nr ) -a A(n +r) = n -2r-no+8
o
B) lle put C(k+l) = c ~ (k+l)
a (k+l) \.
'
s+j
S +j
s+j ) '
lIl
C) lIe introduce n(ks++i> -
Jt - nS+~+k'"
C(k+l) == 0 •
s+j
,
D) "re use each tine the recurrence relation
(k+1)
(k)
(k+1)
- Cs+j-l • Cs+j
CS +j
;
we arrive to find the following recurrence result far k=O,1,2, ... ,d
(75)
~ ~
L
II
n +T-1 ) _
C(k+l) (0
r
• n -1
o
1=0
~
L
(, (
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411
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I
), I
I
(_l)r-s- j (r-s+i)c
_
(_l)r-s- j r-S~t C k+l
j+1 s+j
j+k
s+j·
j=Ii+l
j=Ik+l
:38
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I.
I
I
When
(a"'(n)} -+
we consider
f~r
{(n,n -r) ).
r
]:;:=d
In fact
r
j = I +1=r-s only : hence
d
we have returned to the initial a. b.
I
n(+d~=n
+.
S J
S J
is such that
d
n(d+1)= n
too.
s+j
r,
for
Then C(d+1) = C •
r
r
Therefore we have
+r 1
C =
r
(:0_o 1 -
~+1
LL
d-1
) -
(_l)r-s-
j
e'j:~k)
Cs+j =
k=o j=~:+l
d-l ~+l
n +r-l
\'
\'
( _1)r-s- j r-s+k C
= ( ~o -1 ) _ L
L
(r-s-j) s+j
k=O j=~+l
Since
n
s+j
=n (k+l)= n +r+j+l~
s+j
0
•
far j=I.- +l,I.- +2, •• ,I.- + , lTe find
K
K
-:I.r l
(77)
,.,hich S~lOlTS tha.t
k=k(j)
j=O,1,2, ... ,r-s-l.
and
k(Ik+l)-l:(~)
is a single-valued non-decreasing function of
In fact
= n s +I
k
k(j) = k (constant) for each j=~:+l'''.'~+l
+l-ns+I. - 1 ~
o.
,I{
Then by inverting the subscripts of sununation (k, j) in (76) where
j=O,1,2, .. ,r-s-l, and k=k(j), we obtain, finally, the final recurrence formula
giving tiE number
Cr
I
r-s-l
(78)
n
.-n -s-j
(_l)r-s- j ( S+J 0
r-s-j
) C
j=O
Putting j=r-s-i, i=1,2, .. ,r-s, we have also
s+j·
I
L
r-s
n +r-l
n
-n -r+i
C = (0
)_
(_l)i (r-i 0
) C
i
r-i
r
n -1
o
i=l
Tilis formula not only gives us
J
•
as a function of the r-s-l preced-
C
r
~
rof constructing an automatic method of recurrent calculation "tUch rray be used
ing absorbing coefficients
C i' but also gives us directly a sinple
profitab~r even when an electronic computer is not available.
In fact we can write
n i-n -r+i
)
hr,i= ( r- i 0
n +r-l
r-s
.
C =h O{ 0 1 )- \ ' (-l)\ i C i
r r, n ~
r, r-
uith
i-l
o
and consider the selpence of coefficients
into their components.
Cr decomposed
we find that the terms
Starting from any r
hr,O ' hr+l,l ' hr+2,2' ••• ' hr+i,i••• ·,hs,r_s
are the first
r-s+l binomial
coefficients in the development of the Newton's binomial to the power (n -r-n );
r
0
and such terms repeat themselves for all r' > r such tll8.t n ,-r'=a(n .). n -r.
r
r
r
Renarl~
-
All the problems and results smwn in tlILs wm-k. remain
unchanged when a binary r. w. with a lower a. b. is ccnsidered.
In this case we can
app~
the
e~valence
R.W. [(O,l);(q,p);f{n) < a(n)]E R.W. [(O,l);{q'p');f(n)
where
> ben)]
qt=p, p'=q, g{n):::;n-f{n) (= :frequency of the displacements equal to
f'=O in n steps), ben) = n - a{n).
sen) = n-f{n),
Moreover
gA(n)=n_fA(n)~b(n), b =n -a
r
r
r
=r, etc.
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EX8JIIJ?le
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
a(n)
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.9
5.6
6.3
7.0
7.7
8.4
9.1
9.8
10.5
11.2
11.6
12.0
12.4
12.8
13.2
13.6
14.0
14.0
1(n)
aA(n)
ar
nr
4
4
4
(5)
5
5
(6)
(7)
7
(8)
(9)
(10)
10
4
4
4
4
5
6
0
1
2
5
5
8
9
3
4
7
12
5
10
16
6
12
12
19
20
7
8
22
9
24
25
26
10
r
1
2
3
3
3
3
4
4
4
5
6
6
7
8
9
9
10
11
11
11
12
12
13
13
13
13
(11)
(12)
12
12
(13)
13
(14)
14
14
14
14
14
14
11
12
-------------------------------------------------------------------
41
I
n i-r+1-n
h . = ( r0)
r"J.
1
---------------------------------------------------------------nr -r-n0
i=O
:;
°
°
°1
(0)
4
1
5
:;
6
6
7
8
(~)
8
8
(8)
9
9
(~)
10
10
11
r
4
2
3
(~)
(~)
(~)
(3)
2
(6)
2
(3)
3
(10)
(9)
1
(g)
10
(lg)
(~)
12
10
(10)
(10)
1
(10)
r
a.
(a)
°1
2
°
°
(0)
°
(1)
°
(1)
°
(3)
°
(~)
1
5
6
"
"
a
"
(0)
°
°
(1)
1
(1)
1
(i)
1
(~)
(10)
2
(6)
3
(8)
3
(8)
3
(9)
3
(~)
(~) (6)
5
(~) (8)
5
°
(~)
(a)
1
(~)
r+2
(a)
r+3
3
r+4
(:)
(a)
r+5
5
------------ -----------------------------------------------------------------
(:>
r+a
.----------- -------------.--------------------------------------------------1=0
1
2
4
3
42
5
6
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------------ ----------------------------------------------------------------°
r+1
J
"
"
a
"
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C
r
r
= hr,O(
n +r-J.). r-s
0
_
n -J.
o
L( )
. 1
-1 i h
a
C
r, i r-i
r r
Pr = C;p
q
o
Co
,
= (,)
1
c1
4
= (3)
PI = cJ. P
2
C2
= (~)
P = C P
2
2
J.=
P
o
5
8
1
C5 = (,)+ (J.)C4
8
9
J.O
II
1.2
5 4
4 = C4 p q
P
(i)C,
P = C p7 q5
5
5
J.O 6
P = C p q
C6 = (§)+ (i)C5
6
(,
C7 =(J.o,)+ (J.)C
6- 2)C5
11
8
6
6
P
7
,
P
C8 =( ,)+ (J.)C 7- (2)C6+(,)C5
12
8
8
6
C =( ,)+ (J.)C S- (2)C 7+(,)C 5
9
13
q
P, = c, p5 q'
C4 =
7
q
4 2
4
6
4
P
4
C, = (,)6
(j)+
= C0
9
8
8
a
6
CJ.O=( 3)+ (J.)C 9- (2)C8+(,)C 7-(4)C5
J.4 10
9
8
8
6
Cll=( ,)+( I)CI0-(2)C9+(3)Ca-(4)C7+(5)C6
15 J.O
10
9
8
8
6
C1.2=( ,)+( l)Cll-( 2)C10+(,)C9-(4)C8+(5)C7-(6)C6
6
C pJ2q7
7
= C p 128
q
=
s
P = C p13 q9
9
9
14 10
PJ.O= CJ.oP q
14 11
Pll= CJ.IP q
P12= C~14q12
,--------------------------------------------------------------------------
I
BIBLICGRAPHY
[ 1] WALD, A., Sequential. AnaJ.ysis, New York, John Wiley & Sons, 1~7.
[ 2]
FELtER, W., An Introduction to ProbabUity-Theary and Its Applicatials,
Vol. 1, :New Yorl<:, Jolm Wiley & Sons, 2nd Ed., 1957.
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