I
..I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
•e
I
GEOMETRIC EROODICITY
m
A ClASS
OF DENUMEMBLE MARKOV CHAmS
by
H. D. Miller
University of North Carolina, Chapel Hill
and
Birkbeck College, London
Institute of Statistics M1meo Series No. 431
June 1965
We study the question of geometric ergodicity in a class
of Markov chains on the state space of non-negative integers
for which, apart from a finite number of boundary rows and
columns, the elements Pjk of the one-step transition matrix
are of the form Ck_j where (ck) is a probability distribution
on the set of integers. Such a process may be described as
a general random walk on the non-negative integers with
boundary conditions affecting transition probabilities into
and out of a finite set of boundary states. The imbedded
Markov chains of several non-Markovian queueing processes
are special cases of this form. It is shown that there is an
intimate connection between geometric ergodicity and geometric
bounds on one of the tails of the distribution (Ck). This
connection is explored fully.
This research was sUpPOrted by the office of Naval :Research
Contract No. Nonr-855(09).
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C•
I
I.
I
I
I
I
I
I
I
GEOMETRIC ERGODICTY
m A CLASS
OF DENUMERABLE MARKOV CHAINS
by
H. D. Miller
University of North Carolina, Chapel Hill
and
Birkbeck College, London
1.
Introduction
Consider a hOJOOgeneous" irreducible, aperiodic Markov chain with a
countable number of states identified by the non-negative integers.
the transition probability mtrix by P :: (Pjk)' where Pjk (j,k
I_
f
I
I
I
I
and zero if the chain is null-recurrent or transient.
I
••I
= 0,1,2, ... )
is the one-step transition probability from state j to state k. Let
n
p = (Pjk(n)) be the mtrix of n-fold transition probabilities. It is well
known (see, eg., Chung (3»
I
We denote
lim
n
-+co
that for each j,k the limit
P jk
(n) _
-
7f.
It
exists; this limit is positive for all pairs j and k if the chain is ergodic
The chain is said to be
geometrically ergodic (Kendall (7» if for each pair of states j,k the rate
of approach of Pjk(n) to its limit is geometrically
fas~.
More precisely,
the chain is geoJOOtrically ergodic when numbers M and Pjk exist such that
jk
o .s Mjk < co,
(n) IPjk
~k J
for all Pairs of states j and k.
.s
0
Mjk
.s Pjk < 1
Pn
jk
(n
,
= 0,,1,2, ••• ) ,
(1.1)
Kendall showed that the property of geometric
ergodicity is a class property of an irreducible set of states in the sense
that the geometric rate of approach for one state implies that for all pairs
of states.
MJre precisely again, an irreducible aperiodic Markov chain will
I
I.
I
I
I
I
I
I
I
ae
I
I
I
I
I
I
I
••I
be geometrically ergodic if and only if
Ip(n) _ ~
00
I < M pn
0
for some finite non-negative M and some p satisfying 0
~
p < 1.
State 0
is here meant to represent any given state, the choice of 0 being a matter
of labelling only.
Vere-Jones (13) went further and showed that the rate parameters
Pjk in (1.1) may all be replaced by a single parameter P (o ~ P < 1) uniform
for all pairs of states.
Kendall (8) and Vere-Jones (14) examine the question of geometric
ergodicity for some particular Markov chains, namely the imbedded Markov
chains of certain queueing process.
ing system MlG/l.
lor example, Kendall considers the que-
In this system there is a single server; customers,
arriving in a Poisson process of rate
~
, are served in order of arrival.
The service times of successive customers are independent, identically distributed random variables with distribution function S(.) Which is assumed to
have a finite non-zero mean, conveniently taken to be the unit of time.
=O.
is further assumed that 8(0+)
It
The Poisson rate ~ is in fact the traffic
intensity, i.e., the ratio of the mean service tine to the mean inter-arrival
time.
If we consider the number of customers present (waiting or being
served) immediately after each successive departure then this number forms a
Markov chain for Which the one-step transitim matrix has the form
a
P •
a
l
ao a
l
p, ,a
O
o
p.
•
,0
~
a
2
a,
a,
.~
,a
~
O
a
1
• • •
• • •
• • •
• • •
• • • • • • • • • •
2
,
(1.2)
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
••I
where
an
=
1o
00
n
e -f3x ~ d S(x)
(n
n.
(For a full derivation of this result see Kendall
= 0,1,2, ••• )
•
(5».
All. the imbedded Markov chains considered by Kendall and Vere-Jones
have a property in COJlJD¥:m.
They my all be described as being of the random
walk type, by which we nean that the transition probabilities P
jk are, apart
from a finite number of boundary rows and columns, fUnctions of k-j only.
That is, the P jk are, apart from a finite number of boundary rows and columns,
constant along any one diagonal of P.
clearly has this property.
The matrix (1.2) in the above example
In general these Markov chains are random walks
on the non-negative integers subject to certain boundary conditiona.
The aim of the present paper is to consider the geometric ergodicity
of a random walk on the non-negative integers whose increnents are governed
by a general distribution {c j ; j
= 0,
± 1, ± 2, ••• }. The walk is subject
to boundary conditions affecting one-step transition probabilities into and
out of the finite set of boundary states (0,1, ... ,a).
The one-step transi-
tion mtrix is of the form
P =
Poo
POl
• ••
Pea
Po,a+l
PIO
Pll
• ••
Pw
" .
Pl a+l,
,
Pl ,a+2
Po~
•••••
•••••
• • • • • • • • • • • • • • • • • • • • • • •
• •
• ••
Pro
Pa,a+l
Pa , a+2 •••
Pa+l,o
• ••
Pa+l,
Co
c
P~o
• ••
P~
,
c_l
Co
c
• • • • • • • • • • • • •
c..e
c_ l
Co
Poo
PCXl
"
3
l
c2 ,• ••
l •••
...
(1.3)
I
I.
I
I
I
I
I
I
I
Here P jk =, c k _j for j
> a and k > a, while otherwise the P jk are arbitrary,
but given, and are sUbject of course to the conditions
co
I: Pjk
k=O
=1
(j =
0,1, ••• ,a)
(1.4)
ci
I:
-1= -co
(j > a)
(1.5)
a-j
PjO + Pjl + ... + P ja
I:
The imbedded Markoy chains considered by Kendall and Vere-Jones are particula.r
cases of the above form (1.3).
In addition, the form
(1.3) includes the
imbedded Markov chain of certain many-server queueing processes discussed by
Kendall (6).
As an example we consider one such process in Section
7.
As a starting point we discuss in Section 3 a particular case of (1. 3),
na.m.e1y the random walk on the non-negative integers in which the origin acts
I_
as a natural reflecting barrier. Here the one-step transition matrix is of
I
I
I
I
I
I
I
(3.5) is roughly that geometric ergodicity occurs if and only if there is
••
I
the form (3.5) below.
The main result for a MarkoY chain represented by
some suitable kind of spatial geometric rate of decrease i.e. a geometric
bound on one of the
tails of the distribution (c j ).
This kind of result,
in which, roughly speaking, spatial and temporal geometric bounds imply one
another, has been given for the case of the strong law of large numbers of
Baum, Katz and Read (1).
These authors consider, amongst other questions,
the partial sums 51' 8 , ••• , of a sequence of independent, identically dis2
tributed random variables with cOJllllX)n distribution function F(x) and mean
They show that for given
I,
Pn
=
E ,
the probabilities
S
Pr (
I -; - I.l.I <
S +1
E,
In~1 -
I.l.I <
E ,
•••
)
have a geometric upper bound, i. e. satisfy Pn:S Apn for some A ~ 0,
4
I.l. •
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
,.
I
°s
p
< 1, if and only if the tails of the distribution function F(x) are
exponentie.J.ly bounded, i. e. if and only if
(x > 0)
F(-X) + 1 - F(x) < Be-Ax
> 0,
for same B
" >
°.
In Section 5 we examine the connection between spatial and ten;x>ral
geometric bounds for the mre general process represented by (1.3) and again
we show that this connection is an intimate one.
The question also occurs as
to Whether there is any necessary such oonnecton for general Markov chains.
A simple sufficient condition for geometric ergodicity in a general Markov
chain, given in Section
8, shows that the answer to this question is
in the
negative.
2.
Some preliminary definitions and results.
Let
be a one-dimensional real random variable.
X
Define
X+ = max(O,X) ,
(2.1)
X- = min(o,X) ,
Then X = X+ + X-.
Clearly, if E(
Ix I>
<
co,
then E(X+) <
co
>
and E(X-)
-co •
We say that
E(X) = +
if E(X+) = +
co
and E(X-) >
-co ;
=-
co
and E(X+) <
(2.2)
correspondingly we say that
E(X) = if E(X-)
co
co
(2.3)
co.
We now define a class P of regular functions by saying that a
r~
function belongs to P if and only if its power series has non-negative
coefficients and radius of convergence greater than unity.
We require the following lemma.
5
I
I.
I
I
I
I
I
I
I
I_
~ X1'~'
Lemma A
random variables satisfying Pr(X
1
> 0»
1 < 0)
0, Pr(X
~ Pr(x + ... +X ~ 0) ~ A pn
n
1
>0 2
A
(x
p (0 < P < 1) if and only if
> 0) for some comtants B > 0 2
Proof
(n
_00
For any real numbers
= 1,2, ... )
k ,
1
for some constants
•
> 0 and
+ Xn ~ 0) SApn
Pr(X +
1
> O.
~ E(X ) < 0 ~ Pr(x ~ x) ~ Be-rpc
1
1
>0
T}
Suppose first that for A
0
<p<1
(n = 1,2, ••• ).
(2.4)
, k n +1 satisfying the comition k1 +k2 + ••• +kn+-1 = 0,
we have that
Pr(X1 + ••• + Xn +1 ~ 0) ~ Pr(X1 ~ k1)Pr(~ ~~) ••• Pr(Xn +1 ~ kn +1 ) •
A > 0 such that Pr(X ~ - A ) > P and take
Choose
k_
-"2
= •••
1
= k n+1
=-
A.
Apn+1 ~ Pr(X + ... + Xn + ~ 0) ~ Pr(~ ~ nA)
1
1
{
we have that for
k
1
= n,A
,
Then
and, defining the number
I
I
I
I
I
I
I
I·
I
... be identically distributed,independent
T}
>
(Pr(X1 ~ - A) JO ,
0 by
LIlA
Pr(~ -AlI
e
-T}
,
=
n = 1,2, •••
•
Now for
nA ~ x < (n+1)'A,
Pr(~ ~ x) ~ Pr(~ ~ n'A)
< Ap e -T}nA
< Ap eT}'A since
x-nA < 'A.
Define
T}X
B = ApeT}'A and then we have imependently of n and
6
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
I·
I
for all x
>0
,
It follows now that
E(~) < co.
We cannot have E(X )
1
> 0 for by the weak
law of large numbers this would violate (2.4).
=0
Nor can we have E(JS.)
for then, according to a result of Spitzer
1::!n Pr(Xl
> 0) is divergent
nHence we must have -co:s E(JS.) < 0, and the
«ll), Theorem 4.1), the series
this also violates (2.4).
+ ••• + X
and
necessity part of the lemma. is therefore proved.
To prove the sufficiency part, suppose that
and that for some Calstants B
> 0, 11 > 0
,
Pr(Xl ~ x) :s Be- 11x
Let H(x) be the, distribution function of ~.
l
exists for
t < 11.
(x > 0).
Then the Laplace integral
co t
e x d H(x)
o
1"+(11:S 1"+ :s co) be the abcissa of calvergence of
Let
this integral so that the integral is convergent for t < 1"+ and divergent for
t
> 1"+ (Widder (15». Define 1"_(:S 0) to be the abcissa of convergence of the
integral
fO etx
d H(x)
•
-co
It now follows that ,the. moment. generating function
M(t) =
exists for
f
co
t
(2.6)
e x dH(x)
-co
1"_ < t < 1"+ and so do all its derivatives.
L
co
M"(t)
=
e
tx
x2dH(x)
7
>0
We have that
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
••I
so that M(t)
Since -
is a strictly convex function of t •
S E(JS.) < 0
co
it follows that
-co
< lim M' (t)
- t ... 0+
<0
and so M(t) is decreasing in an interval immediately to the right of t
Hence by convexity there exists a number t
o
= o.
such that
o < to < "t'+
o < M(to ) < 1
M{t) > M(t )
o
(0
Actually, it is clear from convexity that t
,
increasing for to < t <
"t'+
o<t <
= "t'+
Let
<
and to
co
t ~ to) •
is the value of t at which M(t)
or
M(t) is strictly decreasing for
•
H(n) (x) be the distribution function of JS. + ••• + X ' so that
n
(M(t)}n =
Then, since
"t'+ ,
Either M(t) is decreasing
for 0 < t < t 0 and
.
attains its unique minimum.
"t'+
0
St <
t
o
> 0,
J~etxdH(n)(X)
we have
Pr{X + ••• + X > 0)
l
n-
=
1
codH(n) (x)
0-
< lC»exp(t x) dH(n) (x)
o
0-
< (M(t )}n •
o
Since
0 < M(to) < 1, this completes the proof of the lemma.
Corollary.
!.2!:.
Lemma A remains true if we SUbstitute, Pr(Xl +... +Xn
> 0)
Pr(JS.+ ... + Xn ~ 0) ~ Pr(Xl > x) !.2!:. Pr(Xl ~ x).
The proof is exactly the same as that of the lemma. except for the
8
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
I·
I
replacement of •greater than or equals' by
I
greater than' at the appropriate
places.
We shall have occasion to use taboo probabilities (Chung (3)).
H be a given set of states in a homogeneous Markov chain (I: ; n
n
.
Let
= 0,1, ••• )
'Whose state space my be taken to be the non-negative integers.
The n.fold
transition probability of reaching state k from state j under the taboo H
is defined as
HPj(kn)
=
Pr (Y j. H, ••• , Y 1
l
n-
¢
H,Y
n
=kI Y
= j)
,0
(n = 2,3, ... )
where Pjk
3.
is the one-step transition probability for the chain.
Geometric ergodicity in the random walk with a natural reflecting barrier
(process A).
Let
Xl'~'
••• be a sequence of independent, identically distributed,
integer-valued random variables with cOl'lllDOn probability distribution
(c ; k
k
= 0,
±l., :f2, ••• ).
Throughout this paper we make the following assUllIP-
tions about the distribution (c ).
k
Al
The distribution (c ) is strictly two-sided i.e.
k
Pr(XJ. < 0)
>
°,
Pr(X > 0) > 0
l
A2 The set of values of k for which c k
> 0 does
•
(3.1)
no: belOng to the set of
multiples of a fixed integer greater than unity.
Let
Sn
=]L
-"1.
+ ••• + Xn
(n = 1,2, ... )
denote the partial sums of the sequence
9
XJ.'~'
....
Defining So
=
°, we
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
I·
I
rray regard the process (S ; n
n
=. 0,1,2,
••• ) as a free random walk on the one-
dimensional lattice of integers.
In this section we consider in detail, from the point of view of
geometric ergodicty, a new process (Tn; n = 0,1, ... ) defined as follows
T =j>O,
o
Tn
-
=
max[O, Tn-1 + Xn ]
(n = 1,2, ... ),
where j is a given non-negative integer representing the point at which the
process starts.
The process Tn is a randan walk on the non-negative integers
in which the origin acts as a reflecting barrier.
The origin is a natural
barrier in the sense that movement off the barrier is governed essentially
by the distribution (c )
k
and not by some other given boundary coIliition.
For the sake of brevity we refer to the process (Tn) as process A.
Process
A is clearly a Markov chain with transition probabilities
PjO
= q_j
(j = 0,1,2, ... ),
Pjk
= c k _j
(j2= 0, k 2= 1),
(3.3)
where
The matrix of one step transition probabilities is thus of the form
•
,e
,e
• • •
• • •
p =
(3.5)
· ...
,
• • • • • • • • • • • • • • • • •
a particular case of the nx>re general form (1.3).
The assumption Al and A2.
ensure that the process is aperiodic and irreducible.
10
I
I.
I
I
I
I
I
I
I
I_
Let FOO (8) denote the generating function of' f'irst return probabilities
:fb r the origin, that is
,
(:~. 6)
where
f~) = prob
(T
l
~
0, ••• , T _
n l
~
0,
=0
IT0
= 0).
It was shown by KendaJ.l (7) that an irreducible aperiodic Markov chain with
a countable number of' states is geometrically ergodic if and only if the
power series (3.6) has radius of' convergence greater than unity or alternatively
if and only if the functim
1 - s
1 - F
is analytic for
Is I < 1
00
(s)
+ 8, for some 8
> O. Thus according to Kendall's
f'irst cmdition mentioned above, the process (T ) is geometrically ergodic
.
I
I
I
I
I
I
I
I·
I
Tn
n
if and o~ if Foo(s)e P •
Now it follows from a theorem due to Baxter (Spitzer (12), Theorem
3.1) that Foo(S) has the f'ollowing form:
=
F00 (8)
where
S
n
co
sn
.
1 - exp (- Z -Pr(Sn< O)}
n
n=l
( Is I < 1)
is the free partial sum defined at (3.2).
In Spitzer's statement
of Baxter's theorem there is a condition on the first moment of the distribution (c ).
k
However, by a suitable interpretation of the generating
functions involved, this condition can be seen to be unnecessary.
Thus we
can assume (3.9) to hold for a process governed by a quite arbitrary distribution. (c )
k
result.
•
Regarding the nature of process A we have the following
Process A !!.
11
I
I.
I
I
I
I
I
I
I
I_
(i)
ergodic if and onJ,Y if
00
1: 1. Pr(S
n
n
n=1
(1i)
> 0) <
(3.10)
00
transient if and only if
oe
1:
n=l
(iii)
1:
< 0) <
n Pr(S n-
00
(3.11)
;
null-recurrent if and only if the above series (3.10)and (3.U)
are both divergent.
To prove this we let
s -.1- in (3.9) and use Abel's theorem and its
converse for power series "lith non-negative coefficients.
If (3.11) holds
then F oo (l) < 1 which gives transience. If the series (3.11) is divergent
then F 00 (1) = 1 and we have that the process is recurrent. Since for all n,
Pr(Sn< 0) = 1 - Pr(Sn
> 0) and since
00
exp (- 1:
n
s
Ii")
(Is I < 1)
= 1 - s
,
n=l
I
I
I
I
I
I
I
I·
I
it follows that
F
00
=1
(s)
00
n
- (l-s) exp ( 1: !- Pr(S > 0»)
n~ n
n
(3.12)
and hence that
(S)
1 oo
_ s
1-F
00
=
exp (
1:
n
!- Pr(S > o})
n=l n
•
n
Asswning that F (l) = 1, and letting s-.l- in (3.13) we have that the mean
oo
recurrence time is given by
F't (1) = exp(
00
00
1
1: - Pr(S
n=l n
n
a result due to Spitzer «12), p. 158).
therefore follow.
12
> 0»)
,
The results (i), (ii), and (iii)
I
I.
I
I
I
I
I
I
I
I_
It was shown by Spitzer ((ll), Theorem 4.1) that a sufficient
condition for (3.10) to hol~ is that E(
IXl D <
00
and
E(JS) < o.
Using
the definitions (2.2) and (2.3) of infinite expectation, it follows that a
slightly IOOre general condition for (3.10) and hence for process A to be
ergodic is a fortiori that
•
Similarly a sufficient condition for the transience of process A is that
•
We now turn to the question of geometric ergodicity.
We need only deal
with the cases where proceSS A is either ergodic or transient since a nullrecurrent Markov chain, not possessing finite IOOments of recurrence times,
can clearly not be geometrically ergodic.
Note that in our terminology
geometric ergodicity does not imply that the chain is ergodic; it implie s
I
I
I
I
I
I
I
I·
I
that the n-fold transition probabilities converge
limits.
geom:!trica~
fast to their
The limits will be zero in the case of a transient chain.
We now prove the following resuJ.t connecting geometric ergodicity with
the tails of the distribution (c ) •
k
(i)
Necessary and sufficient conditions for process A to be ergodic
and geometrically ergodic are that -
(k = 1,2, ••• ) for sane C
(ii)
00
:s E(~) < 0 and t:tat
> 0 and some }.. (0 < A < 1) •
Necessary and sufficient conditions far process A to be transient
and geome:t~.ically ergodic are that
(k = 1,2, ... ) for same D
0 < E (~) :s
> 0 and sone
~
00
an~ that
(0 < ~ < 1).
We nay express this result alternatively as follows.
(i)
F
(s)e P
00
-
00
ck:s C }..k
and F
(1) = 1
00
~ E(JS.) < 0 ~ ~Ockz
k
E
13
if and only if
P •
k
c -k :s D ~
I
I.
I
I
I
I
I
I
I
I_
(ii)
~
FOO(S)
€
€
~
P •
To prove (i) suppose first that for given C, A (C
-00
> 0, 0 < A < 1),
S E(~) < 0
ck S C
If
(k = 1,2, ... ).
It follows from (3.17) that for same constants
We then have from the Corollary to
and
00
k
> 0 c.1 z
LeIllllB. A
B
>0
and
,,>
0
that there exist constants A > 0
p (0 < P < 1) such tllat
Pr(s > 0) < Apn
n
In fact we may take
A
=1
and
p
= M(to )
where
to
> 0 is the unique value
of t at which the JOOment generating function,
M(t) =
00
~
c
k=-oo
I
I
I
I
I
I
I
I·
I
0 < E(~) S
P and FOO (l) < 1 if !Jld only if
assumes its minimum value.
k
e
kt
,
It follows that the series
convergent and so the process is ergodic.
~(l/n)Pr(Sn >
0) is
Further, the power series
~(sn/n)Pr(Sn > 0) has radius of convergence at least l/M(t ) and hence the
o
function
n
00
(- Z .!... Pr(S > o)}
n
n
n=l
is regular for
0 oS
Is, I ~ l/M(to ).
Kendall's condition (3.8).
Geometric erg~icity now follows from
It follows also from (3.12) that the radius of
convergence of Foo(S) is at least l!M(to ).
Now suppose that process A is ergodic and also geanetrically ergodic.
Then the function
1-- s
1-Foo{s)
14
I
II
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
II
Is I = 1 + 5 for some 5 > 0 and is, clearly, also
is a.nal.ytic in the circle
free of zeros in this circle.
loge 1
i -t,- }=
n
00
-
- 00 8/
is also regular for
Hence the function
Is I < 1
L !L Pr(s > 0)
n=l n
n
Hence
+ 5.
Pr(S > 0)
n
for .some A > 0 and .
p
< Apn
(0 < P < 1).
The conclusions that
_00
~ E(JS.)
<0
and ck ~ C. ",k now follow from the Coroll.ary to Lemma. A and the proof of (i)
is complete.
The proof of (ii) is s:im:Uar:
o < E(JS.)
~
suppose that
01
and
(k = 1,1, ... )
for sane D > 0 and some Il
-~,
-X2, •••
(0 < Il < 1).
By
applying LeJmla A to the sequence
we find that
Pr(S
< 0) < (M(t )}n
n -
-
0
~re
0 < M(t ) < 1 and to < 0 is the unique value of t at which M(t),
o
defined at C~.19), assumes its minimum vaJ:ue. The convergence of the series
L(~/n)Pr{Sn ~
0) and hence the transience of the process now follow.
{3.9},
F00 (s) is regular for at least
.
ric ergodicity.
l/M(t).
o
Is I < l/M{t)
0
From
and
so we have geomet.
Further, the radius of convergence of (3.6) is at least
This completes the proof of (ii)
15
I
I.
I
I
I
I
I
I
I
--I
I
I
I
I
I
I
I·
I
4.
Some further results for process A.
The results of this section are required for the following section.
As usuaJ., 1et
F jk (s)
=
co
(n) n
E f jk S
n..1
(j
Fk,
j,k
= 0,1,2, ••• )
be the generating function of the first passage probabilities
from state j to stake k.
f~~)
(n=1,2, ••• )
Then for process A, the theorem of Baxter which
gives the expression (3.9) aJ.so gives the result
F
10 (S)
=1
- exp ( -
co
n
E !... PreS
n
n
n=1
< o)} •
(4.2)
Note that on the right hand side of (4.2) we have strict inequaJ.ities S < 0,
n
whereas in (3.9) we have S < 0 •
n-
We wish to prove some inequalities concerning the generating functions
FjO(S)
(j. 1,2, ••• ).
Suppose that
co
-
~
<
E kC
k
k= -co
<
(4.3)
0 ,
so that process A is ergodic and geometrically ergodic.
in Section
2, tbat tbe moment generating function
M(t)
=
E
k=
exists for
0 ~ t < 'r+ ('r+ ~
co)
eke
kt
(4.4)
-~
and is strictly decreasing for 0 < t < to '
where, to (::s 'r+) is the unique vaJ.ue of t
value.
It then follows, as
for whi ch M(t) attains its minimum
Consider the equation
sM(t) • 1
(1 _< s < (M(t »-1 , 0 < t < t ) •
o
0
This equation has a unique reaJ. root
t = t (s) which satisfies
1
16
(4.5)
I
.-I
I
I
I
I
I
I
--I
I
I
I
I
I
I,
.I
(4.6)
and t (S) is a strictly increasing,continuous function of
1
We have the following result.
s
with t (l) = 0 •
l
Suppose that the distribution {c }
k
satisfies (4.3), i.e. that process A is ergodic and geometrically ergodic.
Then
(1
F
j
o
< s < (M(t0 )
-
(s) < s + s(s-l)M(t) jt
1-s M(t) e
r 1)
(4.7a)
, .
)
(0 < t < t , 1 < s < (M(t)
-
0
Consider first the proof of the inequality (4.7a).
r 1 ).
We mdify the free
random walk defined by (3.2) in such a way that it starts at j > 0 and that
the states 0, -1, -2, ••• are all made absorbing states.
If
N is the t:ine
to absorption and if Fjo(S) is a generating function for process A, then
(4.8)
An extension of the argument by Miller (10) easUy shows that Wald f S
identity,
holds for
absorption.
O:S t
< to.
Here SN(:S 0) is the state reached at the time of
We now make the substitution
s =
(M(t)
r1
(0 < t < t )
-
0
and obtain
N
jt (s)
E[s exp{t (s)SN}] = e 1
1
Since SN:S 0
and
t (s) ~ 0 we have
1
E(sN) ~E[sN exp{t (s)SN}] = e
1
17
1
(1:S s < (M(t o )}-).
jt (s)
1
(1 < s <(M(t )
-
0
(4.10)
r 1)
(4.11)
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
••
I
vlhich, by virtue of (4.8), gives the result (4.7a).
To prove (4.Th), let
A.
op~~)
denote the taboo probabilities for process
These are clearly identical with the transition probabilities of the
mcxlified random walk with absorbing states defined in the previous paragra.ph.
We clearly have that
(n)
00
Pr(N > n) =
(4.12)
I: oP jk
k=l
and also that
00
( Is I < 1)
Fjo(S) = s + (s-l) I: S~(N > n)
n=l
No't" if
{~n);
•
(4.13)
k = 0, ±l, :1:2, ••• } is the probability distribution of the
unrestricted sum Xl + ... + Xn ' we must have that
(n) < (n)
oP jk - gk-j
(n = 1,2, ••• ; j,k > 0) •
Hence
Pr(N
> n) = ~ P (n) <
k=lo jk -
~
:
<
-
(n)
~-j
k=l
LA
k=l
e
kt
(n)
gk-j
(t ~ 0)
_ jt :
(k-j)t (n)
-e
LAe
gk'
k=l
-J
< e jt ~
In particular we may set
Pr(N
t = t
> n).5 e
k=
>
o
jt
0
e(k-j)t
g~n~
-J
-co
0
= e jt {M(t)}n • (4.14)
and obtain
{M(to)}n
,
(4.15)
from which it follows that the series on the right hand side of (4.13) is
convergent for lsi < {M(to)}-l.
Vere-Jones (13).
This also folloVls from the results of
Hence by analytic continuation (4.13) holds for
18
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
••I
Is I < {M(to )
r1
and in addition we have from (4.14) and (4.15) the desired
inequality (4.7b),
F jO(s)
S
s + (s-l) e
jt
~
sn {M(t»)n
n=l
=s
+ s(s-l)M(t)
l-s M(t)
ejt
We now turn to the case where process A is transient and geometrically
In this case we have
ergodic.
o<
co
1:: kck.:s co
k=-co
1::
c_
Z
k
100
(4.16)
k
€
P
and the moment generating function (4.4) now exists for T < t < 0 (T
-
that t
o
-->
-co )
< t < 0 where t (> T ) is again the
o
0 for which M(t) attains its minimum value. We observe
and is strictly increasing for
unique value of t
-
t
is now negative.
We have the following result.
Suppose that the distribution
{c )
k
satisfies (4.16) i.e. that process A is transient and geamL~rically ergodic.
-
Then
~
Fjo(S) ~ ~ e
jt
To prove this we observe that
fen) < Pr(s < _j)
jo nwhere
Sn
is defined at
(3.2).
(j > 0) ,
Hence
fen) < ~j
(n)
jo - k=oClC) ~
.:s
k=~:
S e
e (j+k)t~n)
(t .:s 0)
kt (n)
= e jt { M(t»)n
ji;:
k~-co
e ~
19
•
I
I.
I
I
I
I
I
I
I
--I
I
I
I
I
I
I
I·
I
On
IlD.lJ.tip~ing by sn (0 < s < (M(t) r l ) and summing over n, we obtain
5. Geometric ergodicity of random
walk with im,posed boundary conditions.
We now consider the question of geometric ergodicity in the
geneml Markov chain with transition matrix (1.3).
are such that the process is irreducible.
this Markov chain process B.
IDgic~,
JIm'e
We assume that the Pjk
For the sake of brevity we call
it would be nore appropriate to
consider first the relation between processes A and B in respect of transience,
recurrence, null-recurrence, etc.
We shall see in this section that processes
A and B are recurrent together or transient together, but we postpone until
the next section the question of ergodicity and null-recurrence.
To distinguish from Process A we use Greek letters for the transition
probabilities, etc., of process B.
(n)
(n)
Thus 1t jk ' CPjk
(n)
' H1tjk denote
respective~
transition, first passage and taboo probabilities for process B while
TI
()
jk s
=
CIO
(n) n
E 1tjk s ,
n=l
~'k (s) =
J
CIO
(n) n
00
(n) n
ECPjk s , _.JIjk(s) = E 1t
S
n=l
Ie
n=lH jk
denote the corresponding generating functions.
Throughout this section H will denote the set of states
Our
CO, 1, ••• ex} •
first task is to obtain some relations between the generating functions
of process A and process B.
We have the following decomposition of the first passage probabilities
cp~~)
based on first returns to states in the set H.
For j = 0,1, ••• ,
(1) _
CPjo
cp
- Pjo
(n) =
1t(n) + ~ n~l
1t(r) cp(n-r)
jo
H jo
k=l r=l H jk
ko
20
(n
= 2,3, ••• ). (5.1)
I
I.
I
I
I
I
I
I
By taking generating functions we obtain the following set of equations
~
00
(s)
~jO(S)
=H'"-oo (S)
_J[
o
+
~
_J[
,p(s)~p
k=l H'"O.n.
.n.O
(s) ,
(Is I < 1)
o
= H YO(S)
+ ~ HITjk(S)~O(S)
k=l
(j = 1, ... ,0;
Is I~ 1)
(5.3)
Consider the row sums of the coefficients Ji!jk(s) on the right hand
side of (5.2) and (5.3).
For j€H ,
00
=
Now for
i
= 0+1,Cl+2,
(P jO + jl + ••• + PjO)S + S ~ P ji HITiH(s)
i:::a+l
(5.4)
•••
I
I_
I
I
I
I
I
I
I
••I
and the generating functions
Itik(s) on the right hand side are now tlDse of
process A since they are based on transitions among the states a+l,Cl+2, •••
only.
Using (1.5) and (3.4) we obtain from (5.5) that
(i > 0).
(5.6)
The right hand side of (5.6) is easily seen to be the generating function
Fi-O:,O(S) of process A.
Hence it follows from (5.4) that
o
~
~ ~"k(s) = (P jO+•• •""Pjo)s + s
k~
~
~ PjiFi-o: o(s)
'
i~l
(j€H)
(5.7)
The relation (5.7) will be our min tool in relating the behavior of process
B to that of process A.
We have the following result.
Process B is ergodic and geometricaJ.ly
ergodic if and only if the following conditions hold:
2l
I
I.
I
I
I
I
I
I
I
(i)
i.e.
(4.3)
(ii)
••I
holds;
the series
00
k
ce
k
I: P1kz , ••• , I: PQk Z
k=O
(5.8)
k=O
all belong to the class P •
To prove this result suppose first that (i) and (ii) hold.
With a
view to writing (5.3) in matrix form we define the matric es
~(s)
n(s)
...
~a(s)
.•
~
•
••
=
~(s)
I_
I
I
I
I
I
I
I
the corresponding process A is ergodic and geometrically ergodic,
•••
,
n(s) =
,
•
•
~(s)
tp(s)
•
Equations (5.3) then become
tp(s)
(I - n( s
i.e.
where I is the unit
process
~
=
that for
n(s) + n(s) tp(s)
») cp ( s)
a x a matrix.
0 <s
~
=
,
n(s ) ,
It follows from the irreducibility of
1
(j, k
and so far
positive.
(j
= 0,1,
€
H)
0 < s < 1 all the elements of the matrices
n(s) and
n(s) are
Since we are assuming process A to be ergodic we have that F jo (1) = 1
... ).
It follcnvs from (5.7) that
22
I
I.
I
I
I
I
I
I
I
a_
I
I
I
I
I
I
I
,.
I
a
E ~jk(1) = 1
k=O
(j,k€ H)
Hence the row sums of II(1) are all strictly less than unity.
Now the max:l.maJ.
positive eigenvalue of a positive matrix does not exceed the maxina1 row sum
of the natrix (see, e.g., Debreu and Herstein (4».
(0 < s ~ 1).
s1ngul&r and, a fortiori, so is I-n(s)
Thus I -n(l) is nonThus we nay write (5.10)
as
~(s)
noes)
where
= (I - n(s)}-1
n(s)
=(
ell
E nn(s» n(s), (0 < s < 1).
n=O
(5.13)
-
is defined to be I.
Now (5.9) may also be written in the form
1 ]
[ ep(s)
•
[1- n(S) :0.. -nts)
0· .. • 0 l [1
- - .. J ep(s)
~
J
..
•
The mtrix
is a stochastic matrix all of whose e1ements, apart from those in the first
row, are positive.
It follows that the matrix (5.15) has a simple eigenvalue
1 corresponding to which there is a column eigenvector all of whose e1enents
are equal and this eigenvector is unique up to a mul.tip1icative constant.
setting
s =1
in (5.14) we Imlst have that
=
i.e.
Thus if we set
that
~
00
On
~jo(1)
[t]
=1
,
(j
= 1, ••• ,a) •
(5.16)
s = 1 in (5.2) and (5.7) it follows from positivity and (5.16)
(1) = 1.
Hence process B is recurrent.
23
To prove that process B
I
~
I
I
I
I
I
I
I
is ergodic and geometricaJ.l.y ergodic we observe that we can, by assumption,
choose
(0 < t < t ) so that for all j
t
o
00
E P.'ke
J
k=O
Now for j
= 0,1,
kt
<
00
EH
•
... , F j (s) is regular in the circle Is I < (M(t )
o .
0
rl
•
By
applying the inequality (4.7b) and taking the value of t for which (5.17)
holds we see that the right hand side of (5.7) is regular for 1 < Is I < (M(t)
"power seriesl
•
and a fortiori for O:s Is I < (M( t)
Since thel coefficients of all the
r
functions in (5.7) are non-negative it follows that each of' the functions
HITjk(S)
(j,k E H) on the left hand side of (5.7) is regular for
o < Is I < (M(t) )-1.
for
Therefore each elelOOnt of the matrix n(s) is regular
O:s lsi < (M(t»-l.
Since I-n(s) is non-singular for
0 <s <1
it
a-
follows by continuity that I-n(s) is non-singular for 0 :s s < 1 +5 , for
I
I
I
I
I
I
I
to the class P.
.e
I
some 5
> o. Thus by (5.13) and non-negativity each element of cp(s) belongs
FinaJ.l.y, by (5.2), 4100 (S)E P and so process B is ergodic
and geometrically ergodic.
Conversely, suppose process B is ergodic and geometrically ergodic.
Then 41 jo (l)
=1
= 0,1,2,...
for j
Hence from (5.2) and (5.3)
ex
E HITjk(l) = 1
(j
€
H)
,
k=O
and so from (5.7)
co
p. +••• 4pjrv + E p, .Fi-a 0(1) = 1
JO
~ i=a+l J~
,
Since
0 < Fjo(l) :s 1
and
~=o Pjk
=1
(j e: H).
(je: H) it follows from the non-
negativity of the Pjk that Fi-a, 0 (1) = 1 for all i and j for which Pj i
(i = CX+l,ex+2, ••• , j
E
H).
That same such i and j exist follows from the
24
>0
rl
I
~
I
I
I
I
I
I
I
aI
I
I
I
I
I
I
,e
I
irreducibUity of process B.
Thus process A is recurrent.
Since process B
is geometricaJ.ly ergodic the left hand side of (5.7) belongs to the class
> ex and j
Choose i
Fi~,O(s) €
r.
H s~ch that Pji
€
r .
> O. From (5.7) it follows that
Now from (4.12) and (4.13) we have
Fjo(S) = S + (S-l)
~
~ op~~»
sn(
n=l
(j = 0,1,2, ••• ).
k=l
Since process A with zero as a taboo state has independent increments we have
that
~
(n)
k=l oPok
<
Thus since Fi-a, o(s)
that F
00
(s)
€
r.
~
(n)
k=l oPjk
€
r
(j
for some
= 1,2, ••• )
> ex it follows from (5.20) and (5.21)
i
Hence process A is geometrically ergodic and since it is
recurrent it is also ergodic.
Further, from the inequality (4.7a), it follows that unless all the
functions (5.8) belong to the class
for some j
€
H, be divergent for
s
r ,
the right hand side of (5.7) would,
> 1, contradicting the fact that the left
hand side of (5.7) belongs to the class P.
The proof of our result is
nOvT
complete.
Next, we deal with the question of transience and geometric ergodicity
and we have the following result.
Process B is transient and geometrically
ergodic if and only if process A is.
The proof of the previous result shows that process B is recurrent if
and only if process A is recurrent.
Thus process B is transient if and only
if process A is transient.
Now suppose process A is transient and geometrically ergodic.
inequality (4.17) to the right hand side of (5.7).
We apply
Since for j = 0,1, •••
F o (s) is regular in the circle lsi = {M(t0 »)-1 and since t < 0 in (4.17)
j
25
I
\.I
I
I
I
I
I
I
aI
I
I
I
I
I
I
.e
I
it follows that the right hand side of (5.7) belongs to
r.
So therefore does
the left hand side and the same argument used for the previous result shows
that ~oo(s) € P.
E;ence process A is transient and geometrically ergodic.
Conversely, suppose that process B is transient and geometrically
ergodic.
Fi-a,o(S)€ P.
for n
> ex,
Then the left hand side of (5.7) belongs to P and so for some i
= i-o:.
For this i, write k
Choose
m so that
a'P~»
Now
O.
>m
fen)
>
ko
-
(m) f(n-m)
J'kl
10
'
from which it follows that F10(S)€
co
r.
Hence from (4.2) we have that
n
!!- Pr(S < 0)
1:
n=l n
€
n
r ,
and so from Lennna. A and from the resuJ ts of Section .3 it follows that process
A is transient and geometrically ergodic.
6. Null-recurrence and ergodicity
in process B
We have seen in Section 5 that processes A and B are recurrent together
or transient together.
However, the relation between these processes in
respect of ergodicity and null-recurrence is a more delicate question.
If we
assume that
co
1:
k=-co
Ik I c k
(6.1)
< co
..
then it is possible to explore fully the relation between these processes.
If ho"rever we assume, say, that process A is ergodic with
(6.2)
in the sense of (2• .3), then it seems that the ergodicity of process B will
depend on that rates at Which
c
k
-4
0
26
as k
-4 -co
and p jk
-4
0 as k
-4
co
I
~
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
for j
€
H.
We shall not explore any further the ini>lication of (6.2) for
ergodicity in process B in the absence of further assumptions.
Suppose
(6.1)
holds.
Then it is necessary and sufficient for the
ergpdicity of process A that
co
=
E(Xl)
t
k c
-eo
k
This follows from the results of Section 3.
of process B when
If
(6.1)
(6.1)
and
(6.3)
and
We investigate now the ergodicity
hold.
hold and if we let E(~,> = -~
(6.3)
(~>
0) then it
follows from a renewal-type theorem of Chow and Robbins «2), Theorem 2), that
,.., -1
]'! (1)
~o
(i
~
~ co)
(6.4)
•
Differentiating (5.7) and setting s = 1 we have
co
t
-J[j'k(l)
k€ H Jr-
=1
+
t Pj.F!.~ (1)
i:::a+1 ~ ~ ......,o
(j
€
H)
Thus provided that
r: ipj' <
~
i=O
we have from
(j€ H)
co
,
(6.6)
(6.4) and (6.5) that
(j
while if for some
j
€
r: ip .. =
i=O
then from some j
€
€
H)
(6.7)
H
J~
,
co
(6.8)
H
= co
(6.9)
•
.e
I
(6.3)
<0
27
I
~
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
Hence if (6.6) holds for all
j € H then for all j, k € H
IfIjk(1) <
co
wh:D.. e if (6. 8) holds for some j
€
(6.10)
,
H, then for some j, k
(6.11)
Tlms if we apply these results to (5.2) and (5.3) after differentiating and
setting s = 1, then we have that process B is ergodic if (6.6) holds for all
j € H and nuJ.l-recurrent if
(6.6) does not hold for all
j
€
H.
Next suppose that (6.1) holds and that
co
E(X1)
=
Z
-
co
k ck
so that process A is nuJ.l-recurrent.
=0
(6.12)
,
Then Fjo(l)
= co
for j
= 0,1, ••••
It follows from (6.5) that (6.11) is true for sane j,k € H and again using (5.2)
and (5.3) we see that process B is null-recurrent.
Again let
(6.1) hold and suppose conversely that process B is ergodic.
Then process A must be ergodic and (6.6) must hold since tIle ergodicity of
process B precludes the transience or null-recurrence of process A.
If (6.1)
holds and process B is null-recurrent then clearly either process A is ergodic
and (6.8) holds for some j
€
H or process A is is nuJ.l-recurrent.
Sunnnarizing, we have the following.
If (6.1) holds then process B is
ergodic i f and only if process A is ergodic and. (6.6) holds for all j€ H;
if ~rocess A is null recurrent
:erocess B is nuJ.l-recurrent/or if process A is ergodic ana: (6.6) does not
hold for all j
€
H.
.e
I
H,
€
28
I
~
I
I
I
I
I
I
7.
An Example
We consider the imbedded Markov chain of the queueing system GI/M/me
(general, independent, identically distributed inter-arrival times, exponential
service tires, m (> 1) servers, na.turaJ.. queue discipline).
distribution function of inter-arrival times and let b
served) of a newly arrived customer at the instant of his arrival forms a
Markov chain with one-step transition matrix of the form
•
Here
~
is an
mxm matrix, and Q is a matrix of the form
4
b
b
l
0
b
b
l
2
b
b
Q4 =
3
•
2
•
0
0
0
·..
b
0
b
l
•
0
0
• ••
b
0
• ••
• • •
0
Apart from the element in the lower left hand corner,
of zeros.
Q4 concerns us.
The non-zero elements of Q4 are given by
b
=
k
fOO
-emu/b)
e
0
k
(mu/b)
k:
d A(u)
It follows from our results that the process is
(i)
~
is composed
entire~
From the point of vievT of geometric ergodicity only the form of
ergodic and geometrically ergodic if
00
-00
i.e•
S Z (-k+l)bk
k=O
<0
00
•e
I
be the mean service
Kenda.ll (5) showed that the number of persons ahead (waiting or being
time.
aI
I
I
I
I
I
I
Let A(u) be the
29
,
•
I
~
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
.e
I
(ii)
transient and eeametrically ergodic if
00
I: kb
k=O 1~
0 < z <1
For
00
< 1 and
k
I: b z
k=O k
€
P
•
we have
B(z)
=
Zb
k=O k
zk
mu(Z-l)/b
00
1
=
e
dA(U) •
0
Hence
1
1
00
o
where
p
u dA(u) = -
p
is the relative traffic intensity, and p = 0 if
f
ud A(u) =
00 •
Hence the process is ergodic and geometrically ergodic if O:S p < 1From the theory of IJX>ment generating
a regular function of
Chapter 7).
for some
For
B(z)
z > 1, i. e.
Z in at least the half-plane
mz < 1
(Lukacs, 1960,
to belong to p the integral (7.1) must converge
A( u) must satisfy
1 - A(u )
for some positive constants
and geometrically ergodic if
8.
functions the integral (7.1) defines
C, c.
p
< Ce -cu
In other words, the process in transient
> 1 and if (7.2) holds.
General Markov chains
In the special class of M:l.rkov chains we have considered so far we have
seen that there is an inevitable connection between geometric ergodicity and
same form of geometric bound on the one-step transition probabilities.
As
part of the general and as yet unsolved problem of determining conditions on
the one-step transition probabilities of a general M:l.rkov chain which will be
necessary and sufficient for geometric ergodicity it is natural to ask whether
there is any similar connection in a general Markov chain.
30
That such a
I
~
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
connection is not in general necessary is shown by the following result.
A sufficient condition for an irreducible aperiodic Markov chain to be
ergodic and geometrically ergodic is that the elements in anyone column
of the one-step transition matrixz apart from the diagonal elementz be bounded
away from zero.
To prove this we label the state corresponding to the given column as
state 0 and assume that the state space is the set of non-negative integers.
With the usual notation for transition probabilities etc., we have the
relation
(n
(8.1)
Now
~p(n)
k=l 0 ok
co
=
1:
k=l
(n-l) (
)
oPok
1 - Pko
By assumption there exists a number p
(n=2,3, ... )
(0 < P < 1) such that for k = 1,2, ••• ,
Hence
~
k=l
0
pen) <
~
(n-l)
ok - P k=loPOk
(n
= 2,3,
••• )
and from repeated applications of this inequality we have that
:
(n) < n
oPok _ P
k=l
L<
Hence from
(n
= 2,3, ... ).
(n
= 2,3, •••
(8.1),
~
r=l
fer) > 1 _ pn
00
-
and ergodicity and geometric ergodicity follow immediately•
•e
I
= 1,2, ••• ).
31
)
I
REFERENCES
I
I
I
I
I
I
I
I
I
I
I
I
I
(1)
Baum, L. E., Katz, Melvin and Read, R. R. Exponential convergence
rates for the law of large numbers. Trans. Amer. Math. Soc.,
102 (1962), 187-199.
(2)
Chow, Y. S. and Robbins, Herbert. A renewal theorem for random
variables which are dependent or non-identically distributed. Ann. Math. Statist., 34 (1963), 390-395.
(3)
Chung, Kai Lai. Markov chains with stationary transition probabilities. (Springer-Verlag; Berlin, 1960).
(4)
Debreu, Gerard and Herstein, I. N. Non-negative square matrices.
Econometrica, ~ (1953), 597-607.
(~
Kendall, David G. Some problems in the theory of queues.
Statist. Soc. Sere B, 13 (1951), 151-173.
(6)
Kendall, David G. Stochastic processes occurring in the theory of
queues and their analysis by the method of the imbedded
Markov chain. Ann. Math. Statist., 24 (1953), 338-354.
(7)
Kendall, David G. Unitary dilations of Markov transition operators
and the corresponding integral representations for transition probability matrices. In U. Grenander, ed., Probability and Statistics (Wiley; New York, 1959).
(8)
Kendall, David G. Geometric ergodicity and the theory of queues.
In Arrow, Kenneth J., Karlin, Samuel and Suppes, Patrick,
ed., Mathematical methods in the social sciences, 1959
(Stanford University Press, 1960).
(9)
Lukacs, Eugene.
(10)
Miller, H. D. A generalization of Wald's identity with applications
to random walks. Ann. Math. Statist., 24 (1961), 549-560.
(11)
Spitzer, Frank. A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc., 82 (1956), 323-339.
(12)
Spitzer, Frank. A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc., 94 (1960), 150-169.
(13)
Vere-Jones, D. Geometric ergodicity in denumerable Markov chains.
Quart. J. Math. Oxford 2nd Ser., 13 (1962), 7-28.
Characteristic Functions (Griffin;
.e
I
32
J. Roy.
London, 1960).
I
~
I
I
I
I
I
I
I
References (cont.)
(14)
Vere-Jones, D. A rate of convergence problem in the theory of
queues. Teor. Veroyat. i ee Prim., ~ (1964), 104-112.
(15)
Widder, D. V. The Laplace Transform (Princeton University Press,
1941) •
--I
I
I
I
I
I
I
.I
33