PRODUCTS OF STOCHASTIC MATRICES AND APPLICATIONS

This paper deals with aspects of the limit behaviour of products of nonidentical finite or countable stochastic matrices (P). Applications n are given to nonhomogeneous Markov models as positive chains, some classes of finite chains considered by Doeblin and weakly ergodic chains.

Let P0,PI,..., be a sequence of finite or countable stochastic matrices, p(n) the (i j) entry of P P p p(m,n) the (i,j) entry of P lJ n m,n m n m,n In the 'homogeneous' case, when P P the classical Markov chains 0 theory provides a detailed analysis of pn: for ergodic chains pn converges as n-,=o, whereas otherwise pnd+r converges as n-oo for some d > and r l,...,d-1.It turns out that in the 'nonhomogeneous' case, when (Pn) are nonidentical lira infn_oP (m'n) > 0 and (m,n) > 0 imply that i,j {pm,n)i,j''/p,j(m,n)} converges as n/oo in a case that may be thought of as aperiodic (m,n) (n) whereas otherwise {Pi,j /P :n >. m} assume a finite numberoflimit points.
and the limit points of {ri,j /P,j :n m+l (m) E (m) (k)/a m) (k) where (k) lim p(m,n) will be identified in terms of i u n-O i (k) u,j n for some sequence of sets {E(k)}.Our results may be understood without n reference to Markov chains, but the proofs will consider a Markov chain {X :n >. m} with finite or countable state spaces assuming a strictly positive n initial probability vector (m) and the one-step transition probability matrices (Pn) n >. m as the starting point.It will turn out that the structure of the tail o-field of {X :n >. m} is crucial for the asymptotic behaviour of (m) results are obtained under the assumption that lim infn_oPi,j > 0.
A particular case is that of convergent {p(m,n)} where lim p(m,n) will be l,J n lJ identified.Then we look at the case of finite S where more powerful results are obtained without any assumption on (P).Further we specialize our H.COHN results to some classes of finite and countable nonhomogeneous chains and explore some connections with the notion of weak ergodicity.
We do not include in this paper specific applications of products of stochastic matrices which seem to be numerous, ranging from demography as shown by Seneta [21], to recent developments in the theory of Markovian random fields assuming phase transitions (see Kemeny et al [12] and Winkler [22]).
Our paper is a streamlined survey of the literature of nonhomogeneous Markov models from the viewpoint of tail o-fields.
2. TAIL -FIELDS Let (,,P) be a probability space and A a set in .We shall say that A is a P-atom/c set of if P(A) > 0 and A does not contain any subset A' with A'g and 0 < P(A') < P(A).A nonatomic set A in is said to be a P-oompetey nonatomio set of C if P(A) > 0 and A does not contain any P-atomic subsets of .It is easy to see that, in general, may be represented as Un=OAn where A 0 is P-completely nonatomic and AI,A 2 are P-atomic sets of C.This representation is unique modulo null probability sets of .Of course, some of {A.}I may be absent If A 0 is present, we shall say that is nonaoo whereas if A 0 is absent is called oio.If A 0 is absent and there is only a finite number of atomic sets {A.} we shall say that is finite.Finally, is said to be trivial if A .
Take now S x S x... where S is finite or countable, Xn() 00n for 0--(I n and write n for the -field generated by {Xk:k > n}.A strictly positive distribution (m) (m) (i ;i g S) and a sequence of S x S stochastic matrices (Pn)n > m uniquely determine a probability measure p(m) on m such that {Xn:n > m} is a nonhomogeneous Markov chain on (, m,P (m)) with p(m)(Xm=i) (m)i and p(m) (Xn+l =j IXn =i) Pi,j(n) for i,j g S and n > m.This model will allow us to use probabilistic arguments on all p(m,n) since l,J the equality p(m,n) p(m)(Xn__JlXm__i makes sense in view of P(m)(Xm=i) > 0 The usual model {X :n > 0} may not always lead to p(O) n (Xm=i) > 0 even if we take (o) to be strictly positive.
n n n n completing the proof.(n)-p(x_=j) for n >.m.We shall say that j probability vector (m) and write j (n) > 0.
is positive if lim infn_o j PROPOSITION 3.1.A state j is positive if and only if for any subsequence (nk) with limn_>oon k there exists a state i (possibly depending on (nk)) (m,n k) such that lim suPk_oPi,j > O.
PROOF.Since !n) (m)p(m,n) and (m) > 0 for i g S it suffices to notice that limk_o j 0 if and only if limk_oPi,j 0 for all i g S. Proposition 3.1 shows that the definition of positivity for j depends only on {P (m,n) ;n > m, i E S}. PROOF.Notice first that {X =j i.o.} e (m) Assume by way of n contradiction that {X =j i.o.} does not equal a finite union of atomic sets n and therefore we may find some infinite sequence of disjoint sets TI,T 2 e (m) with P(m)(T i) > 0 for all i such that {Xn =j i.o.} U T k P(m)a.s.k=l (3.1) (n) Take 6 lim infn_ow.If (3.1) were true, there would exist a set T k with p(m)(Tk) < 6.Let {E (k) :n m} be some subsets of S such that n limn_{Xn eE(k) Tk p(m) a.s. as ensured by Proposition 3.1.In view n E (k) of (3.1) j must belong to an infinity of sets n }" This entails which is absurd.Thus {Xn= j i.oo} consists of a finite number of P -atomic sets of (m).d E (k) for n sufficiently large.
REMARK3.1.Themrem 3.2 as stated as a result about products of stochastic matrices without reference to Markov chains theory.We have seen (m) in the proof that (k) may be expressed in terms of the tail -field u (m) as p(m)(T kIX =u).We shall further consider some characteristic properties of the sets {E (k)} which for some special cases, may lead to the n actual identification of {a (m)(k) }. u THEOREM 3.3.Suppose that j is positive and {Xn =j i.o.} U=ITk p(m) a.s.where {T k} are P (m) -atomic sets of /(m), and let {E (k) be some sets such n that limn-{Xn gE(k)}n Tk p(m) a.s Let F k (t)} be the set of values  (3.12) tends to as n/, it follows that the numerator of Y will tend to as n n/, on a set of positive p(m) probability.Taking account of (3.9) and (3.10) yields p(m)(ngFkP(m)(Anl vine(n)) o) > O, a contradiction that completes the proof.
H. COHN REMARK3.2.For homogeneous Markov chains with period d the sets {E (k) n become {Cu+n(mod d)}, i.e. the cyclically moving subclasses of the recurrent class to which j belongs.In this case p(m) ({Xn g Cu+n (rood d) }IXn-I g Cu+n-I (mod d) }) I, limn_o{XngCu+n(mod d)} T u, u=l d, and (3.7) holds trivially in view (n) of Pj,i 0 for iCu+n+l(mod d)" The results and proofs of this Section rely on Cohn [5], [6] and [8]  We shall next consider a class of stochastic matrices (Pn) satisfying the following condition (m,n) (A) S admits the decomposition S T U C where j T if +/-lmn_ori, j 0 for all m and i and j C if there exists m such that j is positive for {X :n >. m}.
n A partition T, CI,C2,... of S will be said to be a basis for (Pn) if for any j g C k and m large enough, {Xn= j i.o.} =T k a.s. is a P (m) -atomic set of J (m) THEOREM 4.1.Suppose that (P) satisfies condition (A).Then the for i,gS, m 0 and j g C with P,j > 0 is a necessary and sufficient condition for the existence of a basis T, CI,C 2 PROOF.We know from Proposition 3.1 that A= {X =j i.o.} is a n finite union of P(m)-atomic sets of m), i.e.A Uk=ITk.We shall first prove that d=l.Suppose otherwise i.e. d>.2.Since {X =j i.o.}TIUT 2 n Theorem 3.1 implies that there must exist two sequences {n k} and {n} such that (m,n k) (m,n k) lira Pi,j /P,j p(m,n) (m,n) However, .t.mn_ i,j /P was supposed to exist.Thus (4.1 Further, for any set A in (m) with p(m)(A) > 0, the martingale convergence theorem in conjunction with the time reversibility of the Markov property p(m) (AIXn) may be made as close as desired to 1A for n large enough.It is easy to see that if j is positive for {X :n m} it n will stay positive for any {X :n>.m'} with m' >m, and n p(m) (AIXn=J) p(m') (AIXn=J) if p(m) (Xn=J)p(m') (Xn=J) > 0. Thus, one may choose i and m such that for g<1/2 p(m)(T llXm=i) > I-and p(m)(T 21Xm =) > I-g; since T and T 2 are disjoint we also get p(m)(T21Xm=i < e and p(m)(T llXm=) < e.It is easy to see that these four inequalities are in contradiction with (4.2).We have reached a contradiction that proves that A {X =j i.o.} is a P(m)-atomic set of m).
n Conversely, if we assume that A {X =j i.o.} is a P(m)-atomic set of (m) n then Theorem 3 2 yields that limn_oP (re ,j P (AlXm=i)/P (AlXm=) completing the proof THEOREM 4.2.Suppose that (Pn) assumes a basis T, C1,C 2 Then (m,n) (i) for any g Ck, i,Z g S with Z,j > 0 and n large enough k).
jgE n (m) (ii) if for any C k there exists % with 0 < % < such that o (k) < 1-% for JgCk, with k' # k and m large enough, then i gC k implies (n) < oo. and arguing as in the proof of Proposition 1.1.

RE,lARK4
It is easy to see that in case that lira p(m,n) all u g S we get that u (k) limn-ojgCk u,j This happens when (m) limn_oP (X n gT) 0. In particular, the finite chains always satisfy this condition.
The results in this Section were derived in Cohn [8].

CONVERGENT CHAINS.
A chain {X will be said to be ope$ if limn+oop(m'n).(see Maksimov [15], losifescu [ii], and Mukherjea [17]) Of course, this definition depends only on (Pn) and does not need to involve a H. COHN chain {X }.As before our results may be read off without reference to a n Markov chain structure The class of matrices (P)for which {P.(m:n)} convergen is a subclass of that considered in Section4 and, naturally, will lead to stronger properties For convergence chains we shall identify the limits of {p(m,n)} rather than those of their ratios as was done in the previous sections i,J THEOREM 5. I. Suppose that {X is a convergent chain Then (i) there exists a basis {T,CI,C 2 (ii) for any m, igS and j gC k lira P (re,n) that ol > 0, we notice that j/k is also indepedent of PROOF.Theorem 4.1 ensures the existence of a basis {T,CI,C 2 }.
Further, the martingale convergence theorem yields p (m) (Xm=il Xn p (m) (Xm=il m(n_p (m) (Xm=il (m)) as n But _(m,n) (m)/(n) p (m) (Xm=il Xn=J) i, j i j and taking into account that {Xn= j i.o.} T k a.s. is a P(m)-atomic set of (m) we get lira for almost all g T k.Further we can argue as in the proof of Theorem 4.2(i) to conclude (ii) on using (5.1). (ii) yields qij 0 and completes the proof.
REMARK5.2.Theorem 4.2(ii) holds, of course, for convergent chains as wello The additional condition imposed on convergent chains does not seem to make this result any stronger.
The convergent chain concept goes back to Maksimov [15], who considered the case of bistochastic matrices (P).Extensions to finite and countable n convergent chains were given by Mukherjea [16-18], etc.The methods used in these papers are matricial and the sets of the basis are characterized by means of some limit points of matrices {Qk with Qk limn-oPk,n" However, such matricial methods yielded much weaker results and do not seem to permit the identification of {Qk}O The results of this section were derived in Cohn [8].
6. FINITE MARKOV CHAINS: THE GENERAL CASE.
We shall now consider the case when s the number of elements of S is finite.For such chains we shall derive stronger results under less restrictive or no assumptions on (P }.As before we need start off by n considering the structure of (m) (m,m')_(m'.n) Chapman-Kolmogorov formula there must exist g S such that Pi, ,j > 0. Thus ,(n) ,(m')p(m',n) > 0. We have shown that E + '+ Suppose that S is finite.Then there exist some sequences provided that (6.1) (6.2) REMARK.Throughout the paper we will drop the explicit dependence of j upon n and so j is not kept fixed, but in general varying with E (k).The n same convention will be valid for all the sets dependent on n to be further considered.
PROOF.Choose m such that the sets {E+} attached to {X :n m} are n n maximal.Such a choice is always possible in view of Lemma 6.1.
According to (3.5) lie Pi,Xn .,Xne (TklX =i)/e (TklXm=%)  we get that (6 1) holds for states i, j in E + which depend on the m that t was chosen at the beginning of the proof.We shall nest show that (6.1) holds for every t with the same sequences {E_ (k) constructed above for a E + ,+ particular m Indeed assume m' #m.Then with our choice of m E for n > max(re,re'), and ( (m'n') p (n' ,n)/pu( ,nl ,n) + ei,u u,j j (m' ,n)/p (m' ,n) UgEn' i,j E,j u(n,, , (m',n')p(n'.n)/e P,u u, 3 where u' is a state in E + 'n) > 0. Since > 0 the ratio n' with P ,j u p(n',n)/po n).makes sense on the probability space attached to the chain u,3 ,3 {Xn:n m} and (6.5) implies u(n, ',n) (m) (m lira p(n',m)/Pu,j ,j P (T klXn,=u)/P (TklXn,=u') n->o where j g E (k).Using this in (6.6) yields

H. COHN
The tail u-field of a finite Markov chain was proven to be finite in Cohn [2].Further Senchenko [19] and Cohn [3] have independently shown by different methods that the number of atomic sets does not exceed the number of states.
The proof of Theorem 6.1 given here was taken from Cohn [4].losifescu [11]   has studied the tail u-field structure of continuous time Markov processes.
Kingman [13] has given a geometrical representation of the transition matrices of a nonhomogeneous Markov chain from which the tail u-field structure may be derived.
As far as the asymptotic behaviour of transition probabilities is concerned, it seems that the first result in the case of nonasymptotical independent chains was given by Blackwell [i] who derived the existence of limits for the reverse transition probabilities.However Blackwell's paper does not refer to the tail u-field notion.The results of this section on the asymptotics of {P are derived in slightly different forms in Cohn [5] and [7].
m,n 7. SOME CLASSES OF CHAINS CONSIDERED BY DOEBLIN.
In an important but little known paper published in 1937 Doeblin [9] introduced a number of nonhomogeneous Markov chain models and gave without proofs several results concerning their asymptotic behaviour.One model was defined by the following CONDITION (DI).There exists a strictly positive number 6 such that for (n) (n) any fixed states (i,j) either Pi,j >" 6 for all n or Pi,j 0 for all n Doeblin asserted that in this case it is possible to decompose S into disjoint 'final classes' Go,G G v and each final class G, i v may be further decomposed into 'cyclical subclasses' {Ci(); i=l d()}.
These have the following asymptotic properties (according to Doeblln) as (i) e!m:n) 0 for every ieS and jcG O l,j (ii) p(m,n).0 for every i c G and j G (mn) 0 provided (iii) if i j gG with igC() and j C,() then Pi,j that n m # ('-) rood d() (iv) if i j cG with igC() j gC,() then p(m,n) p(n) +gm,n) i,j j ,j provided that n-m ('-) mod d(a).
Here e!m: n) 0 exponentially as n for any m, i and j and the limit 1,3 distribution {p(n)} satisfies jeC,() i",'3 3 i,j where P.3 e.i,3, are as in (iv) and P(m)[i, n ()] is the limit as r of the probability, given X =i, that m Xn+rd( g C().
Doeblin subsequently relaxed assumption (A) allowing positive p(n).to tend to 0 as n oo.More precisely he considered (n) >.
for n >.N or such that for any fixed pair of states (i,j) either Pi,j limn_p (n) 0 Two subclasses of chains satisfying Condition (D 2) were further considered: P (n) those satisfying In=lmax(i,j)gA i,j< (Condition (D)) and those satisfying .n=imax(i,j)gAPi, j (Condition (D)) where A {(i,j):limn_oPi, j To study chains satisfying Condition (D 2) Doeblin proposed introducing an associated chain, derived from the initial one by taking 0 for the positive one-step transition probabilities tending to O.However, by so doing the transition matrices become nonstochastic, and it seems to us that Doeblin intended to add the transition probabilities replaced by 0 to the ones bounded away from 0 in the same row to preserve the stachasticity of the matrix.But there is considerable leeway in defining a matrix in this way and Doeblin's details are rather sketchy.
In the case (DI) it will be easily seen that the associated chain may be defined in anycay described above, but in general we shall have to use some rguments based on the tail o-field structure to justify the definition that we are going to adopt for an associated chain.
We proceed now to define an associated matrix.For the sake of definiteness we shall consider a matrix in which the entries of the initial matrix replaced by 0 are all added to the first entry in their row larger than 6.n) for the pairs row such that (i,j)A and S.I {j:(i,j)EA}, and Pi Pi,j (i,j) such that (i,j) A and j >j(i).
A Markov chain assuming the initial probability vector m) and the transition matrices n'(P'n>.m will be said to be associated to {Xn:nm}.(ii) the sequence tPi,j:i S, n=l,2,...} may contain O's but its positive values are bounded away from O, i.e. there exists 6 >0 such that inf (n) i,j,nPi,j) > 0 where inf' means that the infinum is taken over the strictly positive matrices of the sequence.
Notice that (D)(ii) holds under (DI).As far as (D)(i) is concerned an arbitrary associated matrix P' P n' say may be considered and taking into account only the position of its positive and null entries one may derive the periodicity and the cyclically moving subclasses of a homogeneous Markov chain assuming such type of transition probability matrix.Choosing D to be a multiple of {d(a), = ,v} we can easily conclude that Pm,n+ND have all ND the positive and null entries in the same position as P1 and that e (n, n+ND) 1,3" i,j e Ck(), k=l,...,d()} are all positive for N sufficiently large, 6ND being a lower bound of these entries.Using this we can show that Ck(a fi E* _(n+ND) (n) n Ck() since for j g Ck(), .(ii) If E E .d E(k) n Uk=l n are present then for i eS m=0,1,.., and j g E n lim P(m'n) 0 S. and j(i) also depend on n and should be denoted by S. and j(i,n) re spec t ively.

Let us next consider CONDITION (C).
There exist sequences of disjoint sets {F(): n=O,l n F (e) = ,d' with d' >. 2 and a number N such that _(n) 0 for ie and Pi,j n j F () for n > N = d' n+l Theorem 7.1 shows that (D) is a particular case of (C); the difference between these two conditions lies in that limnooo{Xn e F(n) } Un=N{X noo e Fa(n)} p(m) a.s.need not be an P (m) -atomic set of (m), i.e. it may be a union of P (m) -atomic sets of j(m).
We are now in the position to formulate the following two conditions.
THEOREM 7.2.If (D) holds, then for iS, m=O,l and j g E' n lim p(m,n) 0 (7 6) noo 1,3 and for ieS and j eE (k), k=l,...,d' + o(zj ).(7.7) P i, j j p (m) (Tk) PROOF.It is easy to see that if A {(X n, Xn+N) gB} for B=$X... xs, N+I time s and if ig {j: !m).(m) >0}, then ip(m) (AiXn=i) p, (m) (AlX=i) en (7.8)   n n (i,j)gPi,j for n > m and any N > I.The standard monotone class argument extends (7.8) to any A gn Suppose that we choose an associated chain {X':n u} such that (u) > 0 n i (n) for i E*u and lim infn+oomlniE, i > u Then if we write (7.8) for m u n and ig{j: (u) > O} and take A {X =j} we get j n (u,n) ( ' liS i p,.(u, and therefore J limn_o{X n e E(k)}n Tk p(m) a.s. and (7.8) implies IP(m)( U {XrE(k)}IXm --i)r -p,(m)( U {XrEE(k)}IX'=i) Ir m gm (7 II) r=n r=n and IP(m) {Xr eE(k)}IXm =i)r -p,(m)( {Xr E(k)}iX,=i)ir m gem (7 12) r=n r=n Taking the limit over n yields in (7.11) and (7.12) and using the triangle inequality IP '(m)(lim sup{XeE(k)}IX'=i) _p,(m)(lim inf{XeE(k)}IX'=i) g 2e (7.13) n'-o Multiplying (7.13) by ,.(m) assuming over i and taking the limit over m we get that limn-o{X'-n g E (k) }a.s. with respect to p,(m) exists, has positive n probability and is either P' (m)-atomic or a union of P' (m)-atomic sets of We may now interchange p(m) and P' (m) and get that p (m) (lim sup{X E Ek) }I Xm=i) P (m)(lim inf{X' EE'k)}IXm=i) .<2g (7.14) n n m Now because m) and (m) are finite, we conclude that their atomic sets are in a one-to-one correspondence.Therefore d d' and p(m) (limn-o{Xn e E(k) }n A limn_o{Xn e E' (k') })n 0 p,(m) (limn-o{X'n e E(k) }n A limn_o{X E' (k) })n 0 for k,k' g {I ,d}.Since we have seen in the proof of Theorem 7.1 that E '(k) fl E'* E'* for n large enough it follows that E '(k') c E (k) for k n n n n n sufficiently large.Now it is easy to see that P(m)({X g E' i.o.}) 0 and n n complete the proof by an already familiar reasoning.instead of {E (k) lies in the fact that the former are more easily obtainable.
n For example in case (D) we have seen that such sets may be identified by means of an arbitrary one-step transition probability matrix.
We turn now to the case (D*) which is considerably more complicated than the ones considered so far.We shall first need the following LEMMA 7.2.Suppose that the sequence of sets {A is such that for n m < n <m 2 < n 2< and a sequence {i k} with i kCE k=1,2 and some N>.
(m k ,nk) I pin P. p(m) U (Xn g An}lEa--i) (ii) lip p(m,n)./(n).show that limk_o{Xnk gA a s is a p(m) -atomic set of ,(m), where (m) is (n) the tail o-field of {Xml,Xnl,.... Since Pi,j 0 for igAn and j gAn+1 with n>.N, we get Un=N{Xn EAt lip k-o{XnAnk} p(m) a.s.If limkoo{XnkCAnk} is p (m) atomic (m) not a set of then it must be a union of atomic sets of J' (m) In the latter case, by Proposition 2.1, there must exist sequences A (r) T (r) {A (I) A (v)} with n' c {ml,nl,m2,n 2 n n n' such that lim ,_o{Xn, e n' (say) p(m) a.s.for r=1, v.Further since i keE* there must be a number M such that c U v A (r) for k >.M.But {A (I) A (v)  a.s. is a P(m)-atomic set of ' (m) contradicts (7.15).Hence limk_o{Xnk n k Further limk_o{XnkgAnk} T' a s. is also a Pm)-atomicJ set of --m) since otherwise m) would contain at least two P(m)-atomic sets T and T 2. If {E (I)} and {E (2)} are some sequences corresponding to T and T 2 such that n n limn-{Xn gE(1)}n TI p(m) a.s. and limn+oo{Xn aE(1)}n TI p(m) a.s. and (I)} T p(m) limn-{Xn g E( 2)n T2 p(m) a.s., we also get limkooiXnk g Enk p(m) a.s.we get that i kE* N E (u) for k sufficiently large and (i) follows. ':k=l, To prove (ii) we shall first prove it for a subsequence {n k we shall show that A N E (u)= A for k sufficiently large Notice that n k n k A E is not empty for an infinity of k's.Indeed, by (7.15) and Lemma 7.1 n k n k it is impossible for this intersection to be nonempty for all k sufficiently large, since then (i) would imply the positivity of p(m)(X g A N E i.o.) shall further show that using the existence of such a subsequence {n} we deduce (ii) We first prove (ii) for m >.N and igA Write m Ym, i p(m)(TulXm=i)/p(m)(Tu) and take j A n to get p(m,n) Ym, ir where the prime in the last two sums indicates that the sum is restricted to (nl ,n) the values r such that P > 0. Now if n we may take k and r,j Theorem 6.1 implies that the last sum in (7.16) goes to 0 as k o% We prove now (ii) for arbitrary m and i Consider the conditional probabilities p?(m,r).p(m)(x m'+l Am'+l'''''Xr-I Ar-l'Xr=JlX =i) (7.17) l, 3 m for j EAR, > N, m' max(n-l,m) and r > m' + I, and p,.(m.,m+l) p!m:m+l) Since {Xn gAn _c {Xn+l gAn+l} for n N, a slight modification of a standard reasoning from the theory of homogenoeus chains yields n-I p(m,n) We recall now that for >.N and k gAE we have already shown that p(,n) It follows that for an arbitrary (7.20)Because N' was arbitrarily chosen (7.19) and (7.where the minimum is taken over all (,) (',') 0 for ( a)gk {(,);(',') gx k k=l d}, and that e(,):(,,,) and (E',')k' Then the statement of Theorem 7.3 holds.
We shall omit the proof of this result, which may be carried out by arguments already used in this paper.
We notice that in the case of Condition (D') Doeblin's statement is wrong.
However, examining Doeblin's formulae makes it clear that he felt that unlike the previous situations, the limit of the conditional probability that the chain will circulate through the cyclical subclasses of a fixed class may not exist here.The analogy to the homogeneous case seems to break down for the chains satisfying (D) since several atomic sets of the tail o-field of the associated chain may be lumped into one atomic set of the tail o-field of the original chain.
Theorem 7.3(iv) generalizes a result stated by Doeblin about chains satisfying Condition (D).It is hard to see how Doeblin could have reached his conclusions in this respect, given the knowledge available at the time his paper was written.
The results of this section, in slightly different form, were given in Cohn [7].
One of the main concerns of the theory of finite stochastic matrices has been to characterize sequences of matrices satisfying the so-called 'weak ergodicity' condition, i.e.
(re,n) (m,n) llmri, j ,j 0 (8.1) noo for any i,j, and m This condition has been introduced by Kolmogorov [14] and most papers on nonhomogeneous chains are related to it.Doeblin [9] has found necessary and sufficient conditions for (8.1) and Hajnal [I0] has derived similar conditions unaware of Doeblin's results.We shall first give a result that relates weak ergodicity to the structure of the tail o-field.THEOREM 8.1.The following conditions are equivalent (i) weak ergodicity; (ii) any Markov chain {X :n m} with transition probability (Pn) and n nm arbitrary initial distribution (m) has a P(m)-trivial o-field m).
PROOF.Since S is finite we may assume, if necessary after relabelling the states at successive times n=O,l,..., that there is a positive state j gS.
Then Theorem 3.2 and Remark 3.1 imply that lim pCm,n')/pm,n').p(m)(Tkl X =i)/pCm)CTklXm=A  (8.2).However, if m) is not P (m) -trivial this is not possible as by the martingale convergence theorem p(m) (TklXn--i) must have values close to 0 for some i in view of limn-mP(m)(rklXn IT k p(m) a.s.Thus m) is P(m)-trivial.Suppose now that m) is P(m)-trivial.Then by Theorem 6.2 we know that lid P(m'n)/P(,mn) i. (8.3) for i E But lid p(m) n n-o (X n g E n) 0 for all m and (8.1) follows THEOREM 8.2.Let (Pn) be a sequence of finite stochastic matrices.The following two conditions are equivalent: (i) weak ergodicity; (ii) there exists a sequence of sets {E I)}" such that for i,ES and mgN n (m,n) (m,n) lim ei,j /Pi,j n-co for j g E (I), and for any leS and m=O,l A classical type of results in the theory of nonhomogeneous Markov chains establishes weak ergodiclty in terms of some coefficients attached to a stochastic matrix.A historical account of such coefficients, that goes back to Doeblln, may be found in Seneta [20].Kingman [13] has proven a general result of this kind.Usually, the proof is carried out by some inequalities relating the coefficients of the product of two matrices to the coefficients of the matrices themselves.For example, Hajnal [I0] considered the following coefficient attached to a matrix P with entries P,8 there exists an increasing sequence of positive integers nl,n2,.., such that 'j {Pnj ,nn+l diverges.
PROOF.According to Theorem 8.1, (Pn) is not weakly ergodic if the tail o-field m) is not P(m)-trivial and thus assumes at least two P(m)-atomic sets.According to Lemma 7.1 we must have

j {Pnj ,nj+
There is an important result for bounded positive matrices known in the demographic literature as the Coale-Lopez theorem (see Seneta [21]).The result was given in a somewhat more general form in Seneta [21] and its proof seems rather laborious.The specialization of the Coale-Lopez theorem to the case of stochastic matrices reveals a strong asymptotic independence property We shall state such a property under a less restrictive assumption on the stochastic matrices.THEOREM 8.4.Let (Pn) be a weakly ergodic sequence of stochastic matrices (o,n) such that lim inf max > 0 for any j gS.Then for all i gS n-o igs i,j lim p(m,n)/p(m,n) n_o i,j ,j The proof follows easily from Theorem 3. from being bounded away from 0 for all i,j and n The results of this section were derived in Cohn [5].
p(m)(TklXm u) where T k is an atomic set of the tail m,n u o-field of {X :n >. m}.We first consider the countable case where a number of n (m,n) (m)  -atomic sets of f(m) with d < .

4 .
A CLASS OF COUNTABLE CHAINS.

["
Pj ,i n=l i.C k U T PROOF.Part (i) follows from Theorem 4.1 and Theorem 3.2 if we notice p(m,n) Indeed although it is not SUPn_o[JeCkP(m'n) lin_>oo.m (k) , that p(m)(x n gCk i.o.) p(m)(T k), {Xn EF i.o.} T k (m) P a.s.obtains for any finite set F cC k.This being true for any k and (k) the states of C k being positive necessarily imply that FeE for n large n p-(m:n) (m)(k) which proves (i).enough and therefore limn_o JgUk u,j u For part (ii) we may invoke Theorem 3.3 provided that we show that C kUTE (k) The latter follows from taking E (k) {j:p(m)(TklXn= j) n n noticing that, as shown before, p(m)(Tkl Xn=i) limn_oP (m)(xn gEn(k)}IXn=i)

COROLLARY 5 .
i. Suppose that {X is a countable convergent chain assuming n ,CI,C2,...} and denote qi,k, j EC k (m) (m) (k)/ak and since i g C k PROOF.According to Theorem 5.1, qi,j jai (m) im .> 0 and {X =i i.o.} T k p(m) a.s.we easily conclude that i n p(m) (Tk]Xm=i) as m/oo, and the case i e C k and j g C k follows.Assume now that j ;C k.Then either jeT which yields qij 0 for ieC k, or j C k, with k' #k, in which case ptm)(Tk, iXn=i). 0as n-is a consequence of p(m)(TklXm=i as m+ and T k and Tk, being disjoint.Using now Theorem 5.1

THEOREM 6 .
1.The tail -field (m) of {X :n m} is finite and the n number of Pm)-atomic sets of (m) does not exceed s PROOF.Let T T d be some disjoint sets in m) with p(m)(Tk) > 0 for k=l,...,d.As shown in the proof of Proposition I.I, letting E(k)n {J:P(m)(TklXn=J)>0.5}yields limn_mo{XngE(k)}n Tk p(m) a.s.for k=l,...,d.It is easy to check that {E (k), k=l,...,d} are disjoint for any n n Since p(m)(Tk) > O, {E (k) must be non-empty for n large.However n there are s states in S which requires d s and completes the proof.LEMk 6.1.Let {X :n m} and {X':n m'} be two finite Markov chains n n with strictly positive initial probability vectors (m) and ,(m) and sequences of transition probability matrices (P) (n)i e(m)(Xn=i) (n)m P(m')(X =i)' E+n {i:i > 0}and E '+ {i: (n) > 0}.Then there exists a number N > 0 such that E + E 'and since (m).> O, it follows that p(m,n).> O.By the 1 1,3

E
finiteness of S makes it impossible that E + '+ cE with strict inclusion for n n all m,n and m' > m.Thus there must exist m and N such that for m' Choose now n > N and j gE'+n" Then there is a state igE + such that ,(n) >. ,(N)p(N,n) > 0 Thus p(N,n) > 0 for i g 'n) >. z(N,n) > 0 and E + E '+ obtains for n > N completing the proof S, {E(k)}, k--I S, such that for m--0,1,... n lim p(m,n)./pm, in) (k). (m) /(k) (m)for J gE(k)n k=l d where (k)=u limn-(k) Pu(,m n) n)/p%,j i, gS,m=0,1,... ,r-1 p(m)(TklXm__ i)/P(m)(T klXm_ )i ) (Tk[Xm=i)/p(m) (T k Xm__) < }_ Tk a.s.k Take r=2 V, --1,2,...For each one can find a number re(v) such that E (r) p(m)(T kA U {X g }) 2 -Define now E "k'l E "r'l for m() n < m(+l) and consider the elementary set n n,k properties A A (B C) (A A B) D (A A C) A A (B U C) (6.4)Using (6.4) yields p(m)(T kA U {X E(k)}) p(m)(T kA U ) (T k iXm,=i p (m) (r k IXm,=%) which completes the proof of (6.1).d E(k) i.o.} P(m) a.s.which entailsTo prove (6.2) notice that {X ngDk=l n p(m)(x gE i.o.) 0. This in turn implies lim p(m)(x ngEn 0 (n.)) is an asociated matrix of P if p,.(n. Denote by {E:n=0,1 a sequence of sets with the property infn-ominigE*n)-i > 0 and write E**n S-E*n If {E**}n are present, Suppose that there exists a sequence of positive integers m <n <m 2<n2 < such that u=l P(mu'nu) the argument employed in the proof of Theorem 3.3 we get that P(m)(A u i.o.) > 0 where Au {Xm =i, X m =j}, u=l,2 But u u p(m)(x n =j i.o.) >. p(m)(A u i.o.) > 0 as stated.uINwhat follows we shall consider a condition that contains (D 1).We call this Condition (D)(n) (i)either the {E**} are empty or lim . 0for i E**,

THEOREM 7 . 1 .
Suppose that (D) holds.Then (i) there exists a sequence of disjoint events of S, say E (I) ,E (d) n=0,1, and a positive integer N such that for any m=O,l,

REMARK7 2 .
The sets E '(k) corresponding to the associated chain {X':n m} n n are in general smaller than the sets {E (k)} for which Theorem 7.1 guarantees n the same convergence property (7.7).It is therefore possible that there exists states i E' with i g E (k) for some k and n have the property p(m)(x =i i.o.) 0. The usefulness of using {E '(k)} n u n )(Xnk Jk i.o.) > 0 for JkgAn k k=l2 p(m) (Tu Xm__i) iS m) p (m) (Tu) H. COHN that p(m)(X g F (a) ult.) 0 and the proof is complete.nnAs a corollary to Theorem 7.3 we shall give a result that describes the asymptotic behaviour of a chain that satisfies Condition (D*).COROLLARY 7.1.Suppose that (D*) holds and let ig(,);(E,,a,) maxigc(a), jgCc,(a) Pi,j Zn=Imine(n) holds then p(m)(rklXm=i)/p(m)(Tklxm__) necessarily follows by a consequence of Theorems 6.2 and 8.1.
2 and Remark 3.1 in view of the fact that {E are empty.This clearly implies weak ergodicity, since as seen in the course of the proof of Theorem 8.3, the failure of weak ergodicity prevents (n,n+r o) show next that such results are immediate consequences of the results given in this paper by proving the following theorem due to Hajnal [I0].THEOREM 8.3.A sequence of stochastic matrices (Pn) is weakly ergodic if We