Internat. J. Math. & MatSci.
(1987) 373-380
373
Vol. I0 No. 2
SOME LIMIT THEOREMS FOR RATIOS
OF TRANSITION PROBABILITIES
RITA CHATTOPADHYAY
Department of Mathematics
Eastern Michigan University
Ypsilanti, MI 48197
(Received October 8, 1986)
BSTRACT.
In this paper,
infinite dimensional forward convergent stochastic chains
have been considered in the framework of [I].
The main result of this paper deals
with the observation that the total flow of probability from the C-states to the T-
states is very small compared to that from the T-states to the C-states, if the chain
is observed for a sufficiently long time.
Some examples have been given to justify
the assumptions involved.
KEY WORDS AND PHRASES.
Infinite
stochastic
convergence, basis, C-states,
matrices,
T-States.
1980 AMS SUBJECT CLASSIFICATION CODE.
1.
60J.
Ik’TRODUCTIOM.
(Pn)
Let
(P)
n
Then
and
is
proved
a
in
be a sequence of finite or countably infinite stochastic matrices.
is called a convergent chain iff for each
stochastic
[I] that
of the state space
Here
matrix.
k
>
Pk,n Pk+l Pk+2"’’’Pn
I, lira
n+=o
for k
<
Pk ,n Qk
n.
{T,C I,
for such a chain there exists a unique partition
S
such that for any limit point
Q
of the
Qk
s
exists
It has been
C2,...}
the following
are true:
Qij
O, for all i if jet
0, if i and j belong to different C-classes
Qkj’
if i,j,k belong to the same C-classes.
One of the problems, here, is to identify the T-states and the C-states for infinite
convergent chains.
THF.OREH 1.1.
Let L
T.
Also, the following theorem is known (see [2]) for finite chains.
Let
(Pn)
be a finite convergent chain with basis
Then for each positive integer
m
r.
I im
m
n=k
m
E
n=k
where
(Pn)AB
(Pn)TCL
(mn)nn
iEeA (Pn)ij"
jB
c
0
k
>
I,
{T,C I,C2,...,Cp}.
R. CHATTOPADHYAY
374
The
above
dimensional
theorem
case
and
other
some
have
results
Some
been
extended
examp_les
have
A convergent, forward stochastic chain
(P)
with
modifications.
some
to
the
infinite-
been discussed
to
justify the assumptions involved
2.
PRELININARIES.
ONDITION (U).
(U)
condition
if for each
Sup
ie T
lira
k
j
n
is said to satisy
in each C-class in the basis,
(Qk)ij
0
Qij
This condition yielded a number of finite space like results for infinite chains in
[I].
However, the next few examples demonstrate that Theorem I.I fails to go over
directly to the infinite state space, even for infinite chains which satisfy condition
(U).
(a)
KKA?LE$.
any
L
T
For a convergent, infinite chain with basis
{T,CI,C2,...}
and
the ratio
m
r.
q=k
(Pq)TCL
m
X
q=k
for
k
>
I,
c
(P)EL
q
may not go to zero as
.
m
For, if
0
I/2
1/22
0
I/2
i/2 2
I12
2
first n rows
0
I12
0
0
0
0
then it has been shown in [I], that for each
lira
n/(R)
Pk ,n
0.
1/2
1/22
0
I/2
I/22
The basis of this chain is of course
m
Y.
n=k
(Pn)TC,{ I}
m
E
n=k
(Pn){
’T
c
{T
k,
{1},
C--
{2,3,...}}. But
for all
k
<
m,
SOME LIMIT THEOREMS FOR RATIOS OF TRANSITION PROBABLITIES
It
can
be
IQ k QII
(b)
If
that
seen
this
for all
0
chain
satifies a condition stronger than
(U)
because
k.
This example is given to show that the previous limit may be finite but not zero.
a
k
2
4(k-l),
k
_>
a
with
k
_<
n
1/2 j
(P3n_2)ij
ij
(P3n)ij
{0,1,2
Here, the state space is
lm
n
P
}.
1/2
I/2
0
k,n
0
r.
E
n=3k+l
so that
c
(p)T
n
(Pn)
<
{0}
m-k
m-k
<
i= j
>
i
j
if
j
n and
if
0
if
j
n, i
if
i
n, j
if
i= j
<
k
k+l
if
<
i
<
k
i=j <k
>
k
0
n
It has been shown in [I] that
1/22
1/2 2
Here, T
n--3k+l
3m
let
i=0, j>1
if
I/2 k
(P3n-
< ak+ 1,
if
I-I/2
{0}, C
I, for
k
{1,2
<
}.
But
m
0} ,T e
m
lim
m
n=k
m
l
n=k
(Pn)T e,
O}
I, for all k
(Pn){0} ,r c
so that the limiting value of the above ratio is not zero.
3.
375
MAIN RESULTS.
LKMMA 3.1.
Let
(Pn)
be any chain (finite or infinite).
Also, let
R. CHATTOPADHYAY
376
{Jl
T
and assue that
m P k,
Qk
7.
i+
,
(Pn)LL c
n
0, for all
(Pm,n)ij
lira
(Pn)LL c
7.
ie L
JeL
PROOF.
7.
Suppose
(Pn)LL c <
n
for all
For any
k
.
>
lemma
the
that
Then we have
are stochastic matrices.
where
>
e
Then for given
(Pn)ij
T.
and L
c
is false.
O,
i}
and for all
m
L
Then there exists
k(e)
there exists
T
such that
such that
k (e),
tEL,
Z
n>k
(Pn)LL c <
(Pk ,n
tL
l
ie S
Z
ieL
c
e.
Pk+l Pk+l ,n
tL
(Pk+l)ti (Pk+l ,n )iLc
(Pk+l)ti(Pk+ l,n
< max (Pk+l n)iL c
ieL
+
iL
c
+ l
leL
c
(Pk+l
Pk+l,n
ti
iL
c
(Pk+I)LL c"
maxteL (Pk,n)tLC _< maxteL (Pk+l’n)tLC + (Pk+I)LLC"
Therefore,
Repeating the procedure,
n
tEL
Therefore, for
>
k
E
jeL
<
tL c-
(Pk ,n
max
k(e)
and
(Qk) tj <
c
which is a contradiction.
3.2.
If
nk0
Z
(Pk+j) LLC
j=l
teL,
e
Hence the supposition is false and the result follows.
is an infinite-dimenslonal initial probability row-vector
and
(Pn)
nk0
has all positive entries, then for each
a sequence
of infinite
it follows that for a convergent chain,
Also, for
m
<
n
lira n
JET,
n (J)
>
k
0
choosing
k0 Pk0,m"
If
sufficiently large
0.
and
n+l (i)
w (i)
n
nm
stochastic matrices, then let
n(i)
Wm(i)
Z
ji
n
n-I
Z
E
ji k=m
(j)
(Pn+) ji
n
(i) [I
(P +)ll]
(3.)
n-1
k(j) (Pk+l
ji
Z
k=m
k(1)
[I
(Pk+l
(3.2)
SOME LIMIT THEOREMS FOR RATIOS OF TRANSITION PROBABILITIES
377
The above identities are use to get the following results:,
Let
(Pn)
be a convergent chain with basis
Suppose that
teT.
Then the following hold:
TSEORKM 3.3.
(A)
[,
(i)
(m)
n tt
n=l
(ii)
For jeT,
lira
n+(R)
(B)
l
[I
lim
C
2.
..}.
(Pn)jt
l-(Pn)t t
0
teS
such that
(Pn)tt
n=|
(ii)
C!
]--(R)
Suppose that there is a state
(i)
{T,
(Pn) j t
0,
uniformly for
t.
j
n+- l- (p)
n tt
The n
t T.
(C)
Suppose that there is only one T-state such that
lira
-
n t
(P)
Then
tT iff
So that
0
(A)
3
<
A.
3
)
for each
0
for each
J
t.
ntt
mntt
[I
n=l
PROOF.
<
A. (0
(p)
n+(R)
and
A,
3
Part (i) follows since
(Pk ,n
j
)
t.
> (Pk+l )t t (Pk+2 )t t
(Pn)tt
tt-
n (Pk+m) tt
(Qk)tt > k+l
which means that
E
[I
n;1
.
(Pn)tt
For part (ii), suppose that it is false for some
a positive integer
N
such that for all
n
>
J0gT.
Then there exists 8
N,
(Pn)Jot _> 8[1- (Pn)tt ]"
Then from equation (3.2) for
n
>
m
>
N, we have
n--I
l
n(t)- m(t) _> k--m
[8
k(J0 -k(t)][(l
Fk+l)tt
>
0
and
R. CHATTOPADHYAY
378
which
but
m,n
because
impossible
is
k(JO >
lira
0
part (i) and because the left side goes to zero as
of
and
k(t)
lira
O.
Therefore
the supposition is false
and part (ii) ms proved
(B)
tsT.
Suppose
n
By (I),
Then there exists
>
(t)
r.
<
n
jet
<
(j)
+
jt
<
n+l jt
[I
n
n(t)
(Pn+l)tt
>
n
n+!
>
N
B [1
then we have from above, for
-
+ (t)
(Pn+l )j t
By condition (ii), there exists
max (P
n
> no
implies
0.
(Pn+l) jt
max
jr t
such that
O
(Pn+l)tt]
[!
2
>
2 B
n
n
O
n+l
(t)
(t).
such that
n
>
N
implies
(Pn)tt]
N,
--< n(t) n+l(t)
m
E
which means that
which contradicts
(C)
(Pn+l)tt _< N+l(t) -m+l(t) <
[I
(i N+I
2
This part follows immediately from parts (A) and (B).
BOR 3.4.
(Pn)
Let
Then for all
its basis.
k
>
be a convergent infinite chain with
I,
m
Z
lira
(Pn)BA
n=k
0
m
E
(Pn)AA
n--k
where
A
(finite set)
T,B (finite set)
lim [inf
(Qn) ti >
E
tsT iC
rrm
c
If
n
n
(A)
(A)-
c.
Moreover, if
0
s
then the above result holds for any
FROOF.
T
l
tsA
m(A)_
(t)
n
A
=T
and
n-I
r.
S
E
tA jet q=m
m
and finite
<
q
n, then
(j))Pq+l) Jt
B
=C
{T,CI,C2,...}
as
SOME LIMIT THEOREMS FOR RATIOS OF TRANSITION PROBABILITIES
n-I
r.
E
q
teA q=m
(J)
q--mE jEBE
n-I
E
q(j)(Pq+l )jA
tEA (finite) and
Now, for all
q
Therefore, for
(t) <RE
q(t)(Pq+ l)tA c
jEB (finite) and
and
q
urn(A) > El
n(A)
E
q--m tEA
(j)
>
n-I
E
q--m
q
sufficiently large,
E
sufficiently large and
m
(Pq+l) tj
jt
n-I
>
379
<
m
n,
n-1
(Pq+I)BA EEl
E
q--m
(Pq+I)AA c
Let (m
by any given sequence of positive integers. Then there exists
r
such that for each r
k, there exists
> 0 such that
(3.3)
(pr)
(m
r
>
Pr
r
E
q=k
E
since
(P)BA--<
q
(Pq)AA
q
E/2
E (P
q=k
Pr
E
q=r
and
c
q)AAC
(3.4)
(Pq)^Ac > >
(3.5)
0.
Pr
E
But
lira
r
q=r
(Pq)BA
0
Pr
E
q=r
from equation (3.3).
Hence from (3.4) and (3.5),
(Pq)AA c
for sufficiently large
r, we have
Pr
Z (P)
q BA
q=k
Pr
<
l
q=k
c
(P)AA
q
and the first part of the theorem follows.
We refer the reader to a result obtained in [2] for the proof of the second part.
It has been proved in [2] that for all
E
E
n=l tEA
(P)
n
i E C
it
Using the above result, the proof of the second part easily follows.
L. 3.5.
We now give an example to show that in Theorem 3.4, the condition
that the flow in the denominator is from
A
to
A
c,
is necessary.
R. CHATTOPADHYAY
380
0
I/n
1/n
1-1/n
l-I/n
0
0
0
0
0
0
0
0
0
By Bernstein’s theorem, the products are weakly
ergodic.
Let P
n
(Pk,n+l)il
Z
(Pk,n)is(Pn+l)sl
n+l
(Pk ,n+l )i2
Z
In fact,
(Pk,n)i2
0
as
n
,
for el_ i
(Pk,n)is(Pn+l)s2
(Pk,n)il
+
(P
n+l
(Pk.n) i2
A
{I}, B
n+.
as
So, the chain is strongly ergodic with T
k,n il
c
{2}.
But if
(2}
m
r.
then
n=k
(P)2
n
1, for all
m
Z
n=k
k
<
m.
(p)
n12
Hence, the assertion is true.
I.
MUKHERJEA, A.
2.
MUKHERJEA, A.,
3.
MUKHERJEA, A.,
4.
ISAACSON, D.L. and MADSEN, R.W., Markov Chains: Theory and Applications, J. Wiley,
and NAKASSIS, A., Convergence of Non-homogeneous Stochastic Chains
With Countable States, J. Multivariate Analysis Vol. 16, No.
(1985), 85-117.
NAKASSIS, A. and ISSAACSON, D., Determination of the Basis of a
Non-homogeneous Markov Chain, Statistics and Decision 2 (1984), 363-375.
A New Result on the Convergence of Non-homogeneous Stochastic
Chains, Transactions of the A.M.S. Vol. 262, No. 22 (1980), 505-520.
New Yo rk.