University Microfilms, Inc., Ann Arbor, Michigan LIMIT THEOREMS for NON-NEGATIVE MATRICES

This dissertation has been microfilmed exactly as received 6&-3006

KLIMKO, Lawrence Andrew, 1940- LIMIT THEOREMS FOR NON-NEGATIVE MATRICES.

The Ohio State University, Ph.D., 1967 Mathematics

University Microfilms, Inc., Ann Arbor, Michigan LIMIT THEOREMS FOR NON-NEGATIVE MATRICES

DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Lawrence Andrew Klimko, B.8., M,8,

The Ohio State University 1967

Approved by

Adviser Department cf Mathematics ACKNOWLEDGMENT

The author wishes to express his deepest gratitude to his adviser, Professor Louis Sucheston, for the many long hours which he devoted to the author's mathematical training, for suggesting the topics studied in this dis sertation, and for his patient encouragement throughout this endeavor._ The author also wishes to thank his brother. Dr. Eugene Klimko, for assistance in reading this disserta tion, and his sister, Mrs. Virginia Carney, for typing the manuscript.

11 VITA

December 13, 1940 Born - Youngstown, Ohio 1963 . . . B.S., The Ohio State University, Columbus, Ohio 1963-1967 Teaching Assistant, Department of Mathematics, The Ohio State University, Columbus, Ohio 1965 . . . M.S., The Ohio State University, Columbus, Ohio

MAJOR FIELDS OF STUDY

Probability Theory Professors Louis Sucheston and Jesse Shapiro Real Analysis Professors Robert Helsel and Leroy Meyers Complex Analysis Professor Bogdan Bajsanski Algebra Professor Wolfgang Kappe

111 CONTENTS

ACKNOWLEDGMENT ii VITA ill I. INTRODUCTION 1 II. THE BASIC LIMIT THEOREMS 7 III. COÏWERGENCE TO ZERO: EXTENSION OF A THEOREM OF GREY 18 IV. ON,THE CHACON-ORNSTEIN THEOREM 50 V. THE STRONG RATIO LIMIT PROPERTY 58 REFERENCES 51

IV I. INTRODUCTION

Let T = (t..) , i,j = 0,1,... be a matrix with non- 1J negative entries and denote the entries of T^ by t^• . We assume throughout that T is irreducible: i.e., for each i,j there is an n = n(i,j) such that t?. > 0 . It is easy to show then, as in the Markovian case, that either all of the series t^j converge or all di verge. In the case of divergence T is calledrecurrent. For each i we define dj^ to be the greatest common divisor of the numbers n for which t^^ > 0 . It is easy to show, as in the Markovian case, that d.1 = d. J for all i,j . The common value d of the d^'s is called the period of T ; if d = 1 , T is said to be aperiodic. Given a vector x , its i-th coordinate is denoted either by or by (x)^ . The vectors xT and Tx are defined in the usual way:

(xT)^ = "^ij

(Tx)i = t^^ X. provided that the expressions to the right aremeaningful. We say that the vector x is left-invariant under T (or simply left-invariant) if xT = x . A similar defini tion holds for right-invariance. V/e write x < y to mean x^ < y^ for all i ; we write x < y to mean x < y and X / y . The vector x^ is defined by; 0%*)^ = max(0,x^) . A simple induction shows that for x > 0 ,

(1.1) if xT = X , then xT^ = x for all n , and similarly for right-invariance. Furthermore (1.2) if X > 0 , xT = X , then Xj^ > 0 for all i , and similarly for right-invariance. Indeed, for x. > 0 and for any j , irreducibility 0 implies the existence of an n such that t^ . > 0 . Then ♦ tJ .n ^ . n "o ^o Finally, when no limits appear on a summation sign, the index of summation runs from 0 to oo . Limiting theorems for matrices T are well-known in the case when T is Markovian; i.e., when 2. t . . = 1 tJ for all i ; in fact these theorems constitute a large portion of the theory of Markov chains. Vere-Jones [11] has obtained similar results for the general case by using the same type of argument as in the Markovian case. In chapter II we take a new approach, one which appears to be simpler and more direct, and obtain limiting theorems by the use of Banach limits. In the two succeeding chapters we extend a theorem of Orey [6] concerning the limiting behavior of Markovian matrices and we give a necessary and sufficient condition for the convergence of the ratios f î V where, f and g are in ^ ^ * the space of all absolutely summable sequences, g > 0 , The last chapter contains theorems on the strong ratio limit property for matrices T . We summarize below the relevant facts about Banach limits. An excellent treatment of this material can be found in C9j * A Banach limit L is a linear functional on the space of all bounded sequences with the following properties: i) L(Xj^) > 0 if > 0 (1.5) ii) 1 (1) = 1 iii) where of course l(x^) is L at (x^, x^, ... ) , 1(1) is 1 at (1, 1 , ... ) , and ^(^^.+1^ is 1 at (x^^ Xgt ... ) . We refer to iii) of (1.3) as shift- invariance of 1 . In this thesis all Banach limits are taken with respect to n or N . Existence of Banach limits can be proved by applying the Hahn-Banach Theorem ([2], p. 62) with p(x ) = lim (sup x. .) ; see ^ n — >CD d ■+0 C93. We use the following two theorems on Banach limits. Theorem 1.1. If l(x^) = s for all Banach limits 1 , then (1.4) lim x. , .) = s uniformly in j . n — » 0D If (1.4-) holds for a sequence (x^) , we call s the Lorenta limit of (x^) , and we write L-lim x^ = s . Theorem 1.2. If L-lim x^ = s , and (x^^^ " x^)— > 0 , then x^ converges to s . Theorem 1,1 and its converse are proved in [93• The converse gives an easy proof that

(1.5) if x ^ — > X , then L(x^) = x for all Banach limits L . (1.5) also follows quite easily from the defining properties of Banach limits. Indeed, for s > 0 , choose k such that x^^^ < x + e for n = 0,1, ,,, , Then

L(%n) = - L(x + e) = x + c .

Bince e is arbitrary, L(x^^ < x , The reverse ine quality follows by a similar argument. For a proof of Theorem 1,2 see [53. Since L is a linear functional, it is finitely additive, It is also easy to show that L is (countably) superadditive on positive sequences'; i.e., if x^° > 0 , then (1.6) x%K) > 2^^ L(X]^^) .

Indeed, 2^ x^^ > x^^ for all K ; therefore

L(%k x^^) > L(x^^) for all K , and (1,6) follows.

The following lemma gives conditions under which L is countably additive. Lemma 1.5, Let > 0 , > 0 , a^ < OD and sup b^^ < 00 . Then

L(Zj^ ^k^(^k^^ • Proof. L(Sk aj^V) > 2^ aj^L(b^“ ) follows from (1,6), For e > 0 , there exists K such that

^k ^k^k^ ^k=o ^k^k^ ^ » hence

t(2k ^ Ek.o =k:'(bk°) + s S z% ai^LCb^») 4. E

The result follows since s is arbitrary. Finally we make one additional assumption concerning T , namely that (1,7) sup t^. < 00 for all i,j , n (J Because of the irreducibility assumption, it is sufficient to verify (1,7) for a single pair i^ , . Indeed, suppose that (1,7) holds for some i^ , , Then for any i,j , there exist m and p such that t™ . > 0 and o o' th . > 0 , Also, for any n , J'Jo

hence tJ. < (t? . t^ . )“.-1 sup tJ . n ^0» J'Jo n ^o’^o To see how assumption (1.7) relates to the work of Vere-Jones» let r be the (common) radius of convergence of the series 2 t9. s^ (see [11]). Then under the X J assumption of recurrence, (1.7) is equivalent with the assumption that r = 1 . Indeed, recurrence implies that r < 1 , while (1.7) implies that r > 1 , On the other hand, if sup t^^ = oo , there is an m such that

*00 > 1 C ^ r-1 = lim > 1 . n — > 00 Of course (1.7) involves no essential loss of generality since whenever r > 0 , the study of the behavior of t ? . X J reduces to the study of the behavior of t j where

. II. THE BASIC LIMIT THEOREMS

The method of this chapter consists of taking Banach limits on sequences of entries of T^ or on sequences of ratios of sums of such entries, and showing that all Banach limits agree on each of these sequences. Theorem 1,1 then implies Lorentz convergence of these sequences, and con vergence follows quite easily from this, in one case with the additional assumption of aperiodicity. To show that all Banach limits agree on a sequence, we use the fact that Banach limits yield invariant vectors, and the fol lowing uniqueness theorem on invariant vectors. Theorem 2.1. (uniqueness of invariant vectors). Let T he an irreducible recurrent matrix. Let x > 0 and y > 0 be two left-invariant vectors such that for some

i , X. = y. . Then x = y . The analogous result o o holds for right-invariant vectors. Lemma 2.2. Let T be a recurrent (but not neces sarily irreducible) matrix. Let u > 0 be a vector such that (uT^') . is bounded in N for j = 0,1,... . If u < uT , then u = uT . A similar implication holds for right-invariance. Proof. Note that by recurrence of T , (2 .1) (S_ vT^) . = 0 or 00 U for V > 0 . Indeed, 8

and S^ = 0 or oo by recurrence of T , Now let V = uT - u • Then

n=o vT^ = u # - u "*< uT^ which is bounded at each coordinate j . (2.1) now implies that (2^ vT^). = 0 for each j ; hence

V = vT° = 0 . To prove Theorem 2.1 we set z = x - y . Then + < X ; hence z’^T^ < xT^ = X and (z"^T°’)■^T^: . is bounded for each j (by x. ) . Also Ü d z"** < z'*'T , because if z^^ > 0 , then z-""" = = (zT)^ < (z'*’T)^ , while if < 0 , then z^"*" = 0 which is cer tainly less than or equal to (z'^'T)^^ . Now Lemma 2.2 is applicable with u = z"*" , and implies that z'*'T = z"*" . We have (z.’*’)^ = 0 by assumption; hence o by formula (1.2), z"*" = 0 which implies that x < y . By a symmetric argument y < x ; hence x = y . Theorem 2.1 may be rephrased in the following useful way. If X > 0 and y > 0 are two. left (right) invar iant vectors, then there is a constant c such that y = cx : simply set c = y^/x^ ( x^ > 0 by (1.2) ),

Let = 2^^Q t^j . We now prove a ratio theroem. Theorem 2.3. Let T be an irreducible recurrent matrix satisfying (1.7). Then there exist vectors u > 0 and V > 0 with uT = u and Tv = v such that for all

C2 -2) ^ .

Lemma 2.4-. Let T be a recurrent (but not neces sarily irreducible) matrix, and suppose that w > 0 . If wT < w , then wT = w . Similarly, ifTw < w , then Tw = w . In words, a sub-invariant vector is invariant. Proof. Set V = w - wT . Then

(2.5) vT^ = w - wT^ < w .

Now proceed as in the proof of Lemma 2.2, applying (2.1). To prove the theorem we first note that for i ,j fixed, the sequence of ratios is bounded. Indeed, by irreducibility, there exist integers m and p such that t”. > 0 and t?^ > 0 . Also t^^^+P > t”. tj^ t»„ ; hence t^j < where c = (t^j^ t^^)”^, and therefore

- °®oo + »(m + p)(sup tgg) .

Dividing this by shows that the sequence of ratios

^ i V^oo bounded, since = 1 .

Let L be a Banach limit and set q.1 J. = L(sf./S^1J o o ) , Q = C<îi-î) • Then Q = QT = TQ ; i.e.. 10 ■

(2.A-) Qij = \ ^ik ^kj “ "^ik ^kj ' For

(2.5) qik I'(slk/Sl)tkd 5

^(=k 4 k ^kj/

(S^^/Sq q ) » = (q^j) • By (2.4) the vectors (ôj^jô and (ôj^ô left-invariant. But q^^ = q^^ = 1 ; hence by the uniqueness theorem q' V J • = qj! O J. for all j . Also, for each j , the vectors and (q^j)ô are right-invariant. But q . = q^. implies that q. . =

q(.1 J. for all i • Thus v;e have shovm that for i, j fixed, all Banach limits on the sequence agree, and we have that this sequence converges Lorentz. To obtain convergence, we observe that differences of successive terms tend to zero and apply Theorem 1.2 : 11

iqN ^ .N+1„N oN qN oN ^.N+11 leN+l/oN+1 oN /oN I _ ôo”î.i®oo -î.roo ' /®oo - îo'/ôo’ “ ------: 00 00

where K, = sup . and K = sup • This last expres- X ^ ^ oo Sion, however, tends to zero with N , since by recurrence, tends to infinity as N tends to infinity, Next we set v = (Vj_) where v^ = • Since = 1 , (1.2) implies that Vj^> 0 for all i . For each j , the vector I’ight invariant; hence there is a constant u. such that q.. = u.q. = 0 10 0 10 V. u. . Clearly the vector u = (u.) is left invariant. 1 J d Finally we use the decomposition

to complete the proof. Now that we have a right-invariant vector for T , we can apply the well-known transformation p 9 . = t^.v./v. I J ^d d ^ and use limit theorems known for Markovian matrices to obtain the corresponding limit theorems for T . To complete this development we present alternate proofs, using Banach limits, of the limit theorems for Markovian matrices. 12

Theorem 2.5» Let P be an irreducible Markovian matrix. Then there exists a left-invariant vector tx > 0 such that for all i,j

(2.5) L-lim p ? . = TC, n — >00 ^ I.e. t (2 .6^ ) lim = 71. uniformly in m . N->co ^

Either tx = 0 or 2. t x . = 1 . If there is a probability J J vector u ( u > 0 , 2. u. = 1 ) with u = uP , then U J \I — TC *

Proof. Let L be a Banach limit and set q .. = L(p^.) , Q = (q..) . The following lemma holds. -LJ -LJ Lemma 2.6. If P is an irreducible recurrent Markovian matrix, then Q = QP = PQ ; i.e.,

(2.7) ^ik^kj " ^k ^ik^ko * Also 2j q^j < 1 .

Proof.

Qik^kj ' ^ ^ik^kj ^ “

L(p?th . . where the inequality results from the superadditivity of L , and the last equality from shift-invariance. This proves that the vector (q^ ^) is left-subinvariant; hence 10 0 by Lemma 2.4 it is left-invariant. Q = PQ follows by a 15 similar argument. Finally

^ij " Pid^ " L-(l) = 1 .

To prove (2,5) of Theorem 2.5 we may assume that P is recurrent. (Otherwise •— > 0 as n — > oo for all 1 J i, d and we may take n = 0 .) Let be another Banach limit and' set . = L'(p?\) , Q"* = (q.^^) • By ^ J J- (j 1 o Lemma 2.5 and (1.1),

°-id " ^ik^kd • Taking L' of both sides and using Lemma 1.3 with a^ = and b^^ = p^j yields

^id “ ^k ^ik^kd ’ i.e., (2.8) Q = QQ' . Next,

^id " ^ik^kd and taking L of both sides yields

q'id = L(2% Pik^kd ^ - ^k ^ik^kd "

Hence > QQ^ • Using this and (2.8), we have that Q < . By symmetry Q^<- Q ; thus Q = Q% This means that all Banach. limits on each of the sequences (p?-?) X J agree. 14

îfov/ by (2 .7), for each j , the vector (^ij is right-invariant under P • Hence, sinae (1, 1, ...) is right-invariant under P , and by Theorem 2.1,

(2 .9) = q^j for all i, j, k .

We set T:. = q. . (for any i ) to obtain (2.6). Now J suppose that > 0 for some s . Then by letting L = we have from (2,8) that Q = . Thus

^os ” ^^ok'^ks * But by (2.9) q]_,g. = q^g for all k . Hence

% 8 “ ^os ^k ^ok and dividing both sides by q^„ (which is positive since

% 8 = ’'s > ° ^ G i v e s 1 . Sj. 5^1. = 7U;, .

To prove the last statement of Theorem 2.5» suppose that u is a probability vector with uP = u . Then by (1.1)

Pkj * Taking L of both sides, and using Lemma 1.3» we have

"c = "k^kj ' "j^k “k = ’'J • This completes the proof of Theorem 2.5. The following theorem due to A. N. Kolmogorov asserts 15 that in the aperiodic case there is even unaveraged convergence of Ij . as n — > co . Theorem 2.7. Let P be an irreducible recurrent aperiodic Markovian matrix. Then p?. — > tt. (n — > oo) J. Proof. The proof makes use of the following lemma. Lemma 2.8. Let P be an irreducible recurrent Mar kovian matrix. Then lim sup p^. does not depend on i . 00

Proof. Let j be fixed and set r, = lim sup pP. , ^ n — > CO r = (r^) . For given i and e > 0 , there is a K such that p^^ < e . Then

n — > 00 ^

lij^up PikPgj =

lim sup(p”^ - Pik^kj ^ - r — E n — > 00 Since s is arbitrary, it follov/s that Pr > r • Clearly (P^r) . < 1 since r . < 1 for all j ; hence by Lemma d Ü 2.2, Pr=r • Since (1, 1, ... ) is right invariant for P , it follows from Theorem 2.1 that the entries in r are identical. To prove Theorem 2.7, let i and j be fixed, and n let lim sup pI .. = s . Then by the previous lemma, n■n ---— >^ 00m 16

n * lim sup p,_. = s for ail k . Suppose now that p.. > 0 . n — > 00 There exists a subsequence n such that p?^ — > s Ï1 —1 (n — >oo) • We show that p^." — > s (n — > oo) . .j Indeed, otherwise there is a subsequence n of n^ and an s'' such that p^."^ < s^^ < s for all n . Set Xj e = (s - s'^)pj^j^/$ . There is a K > i such that ^ik ^ • ^ben for all n sufficiently large Pij = ^k PikPkd""^ < PikPkj"^ + ^

îiîj ^ ^k=o îk^® + ^ < k?4i

p^^s' + (l — ( s + e) + £ <

s + 2e - Pj^j^Cs - s") - s + 2E-5e = s- e n" which is a contradiction since p .^ — > s (n — > oo) • / u Hencep^ — > s (n — > oo) . Repeating the argument, / we have that p? — > s (n — > oo) for d = 0 , 1 , . X J In case _p^^ = 0 , aperiodicity implies the existence of an integer m such that p?f^ > 0 for n = 0 , 1, ... . The previous argument then gives that > s (n — > 00) for d = 0 , 1, ,,, . This, together with (2.6), i.e., Lorentz convergence, implies that s = rc. . J A similar argument can be applied to show that

lim inf p ? . = t c . ; hence lim p^. = u . . n — >00 ^ ^ n — >00 ^ ^ 17

[Je state the analogue of Theorem 2.7 for (not neces sarily Markovian) matrices T , It can be proved, of course, by transforming via . = t^.v./v. to the Mar- J- J X J J X kovian case, and using Theorems 2,5 and 2.7. Theorem 2.9. Let T be an irreducible recurrent aperiodic matrix satisfying (1.7), and let v be the non negative right-invariant vector for which v^ = 1 . Then there exists a vector u > 0 with uT = u and

lim t ? . = v.u_. . n —> 00 ^ ^ Either u = 0 , or 2^^ ^i"'^i = 1 •

In the event that 2^ ^i''^i = 1 * are positive for all i and T is called positive recurrent, other wise T is called null-recurrent. III. ^ CONVERGENCE TO ZERO; EXTENSION OF ^ A THEOREM OF OREY

In 1962 Orey [6] proved a striking result about Markovian matrices; namely, that if P is an irreducible recurrent aperiodic Markovian matrix and if x and y are two probability vectors, then II ( x - y ) P ^ I I — > 0 (n > co) where II x II = S^lx^l • This result may be stated in the equivalent form: if f e and f = 0 then II fP^ 11 — > 0 (n — > œ) . The transformation p^. = i J t^.v^/v. mentioned in the previous chapter allov/s us to ^ J 0 ^ draw a similar conclusion for an irreducible recurrent aperiodic matrix T satisfying (1.7), provided we make the additional assumption that the positive right-invariant vector V (which exists by Theorem 2.5 and is unique modulo multiplication by a constant by Theorem 2.1) is bounded away from zero; i.e., that there is a number c such that

(5.1) > o > 0

for all i . (Throughout this chapter T is assumed to be an irreducible recurrent matrix satisfying (1.?), and v designates the non-negative right-invariant vector for which Vq = 1 .) Suppose then, that T is also aperiodic and that v satisfies (5.1). If f is a vector such that fv e where (fv)^ is by definition f^v^^ , we have

18 19

(3.2) Il fIl = =

LjlEi f.ViP^^Vjl <

o“^2 .lr,f.v.p?.l = o“^ Il (J -L X ± XJ

'Thus if = 0 , the last expression in (3.2) tends to zero as n tends to infinity by the theorem of Orey mentioned above, and we have IIfT^II — > 0 (n— > oo) . One might expect to draw the same conclusion without (3.1) $ but the follov;ing example shows that this is not

always possible. Let T = (t..)X(J , i, 3 = 0, 1, ... where j if 0 = 0 t.. = i' if = i + 13 I 1 3 1 0 otherwise . Since T is the transpose of an irreducible positive recurrent aperiodic Markovian matrix, T itself is irreducible recurrent aperiodic and -satisfies (I.7). A right-invariant vector is given by v^ = 2”^ . Finally for f = (1/2 , -1, 0 , 0 , ...) , it is clear that fv e ^ 1 and it is easy to verify that

r 1/2 if 3 = n (fT^)j = -’ -1 if 3 = n + 1 I 0 otherwise , 20 so that II f I I = 5/2 for all n . The difficulty arises because the powers of T are in some sense getting large, and the obvious remedy is to place some restriction on the growth of T^ . We will use the following condition:

(5*5) sup IIT^ II < 00 n where IIT^ II = sup Z . t^. is the norm of as an i ^ operator. Clearly (5.3) implies (1,7)* Under condition (5*3) and the assumption that T is positive recurrent we can extend Orey’s theorem. In case T is null- recurrent and satisfies (3.5) but not (5.1) we do not know if Orey's theorem extends. In considering the ex tension of Grey’s theorem under condition (5*3)» the following theorem shows that we may also assume that the right-invariant vector v is bounded; i.e., that sup < CO . The proof of this theorem is based on an idea which Sucheston [10] used to obtain invariant func tions. Theorem 5.1. Let T be an irreducible recurrent matrix satisfying (5*3)* If the right-invariant vector v is not bounded, then for all f in , IIfT^II > 0 (n — > co) . 21

Proof. We show that 2 . t? . — > 0 (n — > oo) for all i • Prom this it follows that for f e ,

(3-4) ' ZjIZl < Si Ifil Sj t“ . , and the last expression in ($.4) tendsto zero with n since 2 . I f. I < œ , sup 2 . t? . < oo and 2 . t ? . — > 0 ^ ^ ^21 o d id (n — > oo) for all i . Let e^ = lim sup 2 . t 9 . . Then e = (e.) is a ^ n — > 00 ^ ^ right-invariant vector. Indeed, for i fixed, let s be positive and choose K such that 2^^ t < e/M where M = sup IIT^ II . Then

li^T^ ^d^k=o ^ik^kj -

lin^ig (2j t^tl _ t,^tgj) >

Since £ is arbitrary, we have that Te > e . Also since e^ < M for all i ,

= Zj t“ .e. < < y2 i thus Lemma 2.2 implies that e is right-invariant. However since e is bounded and v unbounded, the uniqueness theorem (Theorem 2.1) implies that e = 0 , 22

Let T be an irreducible recurrent matrix satisfy ing' (3 «3)t let the right invariant vector v be bounded. We say that T has -nrorertv 0 if for e v e r y f e witü 2^f^v^ = 0 , 0 (n — > oo) . The following theorem gives several conditions which are equivalent to property 0 and which are easier to v/ork with. Theorem 3.2. Let T be an irreducible recurrent matrix satisfying (3.3)* and suppose that the right- invariant vector V is bounded. The following are equivalent.

i) T has property 0 , ii) There exists an a such that

I 0 (^ — > œ) .

iii) Z. It^t^ - t^.\ 0 (n— >oo) for all i . 0 1J — J Proof. i) implies ii). Let f = (fj_) where fi = (Kronecker 6 ). Then g = fT - f e and

SjV. = = Zj = 0 .

Hence IIgT^II — > 0 by assumption. But

IIgT^II = IIfT^^^ - fT^li = S. It^l"^ - t ^ J .

ii) implies iii). Suppose there exists an a such 23 that u cij - t^. aj I — > 0 (n — > oo ) . Let e. X = lira sup 2. I t^t^ - t ? . I , e = (e. ) • V/e proceed as in n — > 00 ^ the proof of Theorem 3*1 to obtain that Te = e ; hence, since e^ = 0 and by (1.2), e^ = 0 for all i . Indeed, let M = sup llT^II , let i be fixed, let e n be positive and choose K such that 2^^ t < e . Then " (lim sup - n — > 00

lim sup (2, 2. t I tPt^ - tîî . 1 - 2eM) > n — » 00 ^ ^ lim sup (2.l2.„ t (t^+^ - tg.)l ) - 2eM = n — > CD J Kj -Ü

lim sup 2. - t?i^! - 2eM = e, - 2eM ; n — >00 ^ ^ hence Te > e . Clearly (T^e) . is bounded (by 2M^); U hence by Lemma 2.2, Te = e . iii) implies i). ]?or b fixed, let be equal to

(3.5) lim sup 2.1v,t^. - v t? . I , n — > 00 ^ ° a DO e = (Oj^) . Note that e.^ = 0 , We use the same type of argument as in the previous implication (making use here of the fact that v is bounded) to obtain that Te = e , and finally that (3.3) is zero for all a, b . The desired implication is then obtained by the following standard 24 approximation. Suppose that f e K and S. f.v. = 0 . U V U If f has at most two non-zero entries, i) follows from (5 .5 ). Assume that i) holds if f has at most n (> 2) non-zero entries and suppose that S e , 2. g -v. = 0 -i- O O d and that g has exactly n + 1 non-zero entries, We may assume that g^ 0 , ^ h^ = g^ , h^ = -SqV^/v3_ , hj = 0 for j > 2 , h = (h^) , Then g - h has at most n non-zero entries and 2. h.v. = 0 , 0 0 0 2 . (s - h) .V . = 0 , But O o U n I II ST^II < ||(s-h)T^|| + II hT

and each term on the right tends to zero by the induction hypothesis. Hence i) holds if f has finitely many non zero entries, How for any f s K with 2. f . v . = 0 , let e be J J 0 positive and choose K such that |2^_ f.v.| < e and J —O J J Ifjl < G , Set G = (f^, f-^, 0, 0, ...) ,

h “ ^do^Lo Vk ' h (s-h)jVj = 0

since = 1 , Also l|hl| < e ; hence

(3.6) IlfT^II < ||(f-g)T^|| + |i(s-h)T^|| +

IlhT^ll < 2s(sup ilT^II) + II (g-h)T^^I! n v/hich completes the proof since the last terra in (3.6) tends to zero as ntends to infinity, and e is arbitrary, 25

The following theorem, which has interest of its own, provides the key for extending Orey’s theorem in the positive recurrent case. Theorem ^.5. Let T be an irreducible, positive recurrent, aperiodic matrix satisfying ( 5 * 5 ) » Let q.. = lim t^. . Then lim Z.t^. exists and is n — >00 n — >00 ^

‘lid Proof. By Patou’s lemma

(5.7) 2. q .. = Z . lim inf t?. < lim inf 2. tj. . d d ^— > CO ^J ^ n 00 d Id

Let M = supllT^ll , = 2. q. • , b. = lim sup 2. t^. • n d Id 1 ^ — > 00 d ^d

Suppose that < b^ for some particular r . Then there exists an > 0 and a subsequence n' such that 2. t^. > a + 2s . By Theorem 2.9, there exists J ^ O u = (u^) such that u = uT and q.- = v.u. ; hence V . 1 J 1 J

2. q. . = V. 2. u. < M , which implies that 2. u- < oo ; u d d d d thus there exists a K such that

(3.8) ^ V o •

Choose N = n'' such that o 26

Then ,K which implies that

“d=K ^rj ^ • Now since u = uT , we have

hence

^rj ^ ^r‘^o which contradicts (3.8); hence a^ > for all i f i.e.; lira sup 2. t?, < 2. q . . n — >00 0 io d ij which together with (5*7) implies the desired conclusion, Corollary 3.4-. Let T be an irreducible, positive recurrent, aperiodic matrix satisfying (3.3). Then T has property 0 , Proof. We note first that by the hypotheses, the right-invariant vector v must be bounded, because it follows from the proof of Theorem 5»1 that the vector e = (e.) , where e. = lim 2. t^_. , is right-invariant, ^ n —> 00 ^ Also, by the assumption of positive recurrence, e is positive, and by (3.3)$ e is bounded. 27

To prove the corollary, v/e show iii) of Theorem 5 .2 . For e > 0 and i fixed, choose K such that q^j < e . Choose N such that

and

whenever n > IT • Then

(5.9) lt-1 _ +

y CD , n+1 ^ y 00 u_n j=K+l ^ij ^j=K+l ^ij ’ and for n > IT , the first term on the rrght of (3.9) is less than 2e , while the last two terms are each less than 3s . Before closing this chapter we present tv/o examples; in the first example T is an irreducible positive recur rent aperiodic matrix satisfying (3 .3 ) but not (5.1); in the second example T is an irreducible null-recurrent aperiodic matrix satisfying (3.3) but not (3.1) and has property 0. Example 1. Let t; . = (5/^) 2~^’"^ • Then T^ = T u. I j for all n and a right-invariant vector v is given by; . The following considerations show some of the difficulty involved in constructing an irreducible null- recurrent matrix T satisfying (3.5) but not (3.1). Let T be such a matrix, and suppose that e = (e^) is a non-negative right-invariant vector with e^ — > 0 (i— > 00) . Then e must be zero. Thus we must construct an example in which lim inf v. = 0 i - ^ c o ^ but lim sup V. > 0 . To prove that e = 0 , let s 00 ^ be positive. Then for each i

®i = ^ ^j=o ^ij where K is chosen so that e . < e for j > K , and J M = sup IIT^II . Since T is null-recurrent, t? .— > 0 n for all i, j as n — > co ; hence e^^ < Ms . But since e is arbitrary, e^ must be zero.

Examnle 2. We define a Markov chain P with state space all integers as follows: p(0,0) = p(0,l) = 1/2 , p(-i,0) = 1 for i = 1, 2, ... , p(i,i + 1) = (i + l)/(i + 2) for i = 1, 2, ... , p(i,-i) = 1/(1 + 2) for i = 1, 2, ... . Clearly P is irreducible recurrent aperiodic. Also P is null-recurrent since 29

2n“ i nf^(0,0) = 1/2 + I/5 = 00 . Now let = 1 for i > 0 , = l/(2-i) for i < 0 , and set *ij = "here p.. = p(i.j) , T = (t^j) .

Then t ? . = v. p?./v. , and 3cJ J- i #J Jel - ^j>o Pij +

^d■ 2j

"0 = A+ + A- where

= 2 .:^_ It^i^ - t^^ d>o 'hj 1 4-ïi-î-l j<0 *oa n-kl n But = ip:/-p:p < =j Ip#' - p:ji -> 0 oj - Poj by Orey’s theorem, while

= ^0^0 0 -p#'

=j |p:j - p:3'i -> ° again by Orey’s theorem*, hence by Theorem 3.2, T has property Q. IV. ON THE CHACON-OHNSTEIN THEOREM

\‘le introduce some additional notation for this chapter. If f and g are vectors, we set

%(f»s) = (^n=o provided that this expression is meaningful; D..(f,g). is the j-th co- C ordinate of the vector D~-(f,g) . V/e write ; f for S . f . • For vectors f and v , fv has the same O Ü meaning as in the previous chapter; namely, the vector whose j-th coordinate is f-v. . Throughout this chapter u d T is assumed to be an irreducible recurrent matrix satisfying (1.7); v denotes the non-negative right- invariant vector for which v^ = 1 . The Chacon-Ornstein theorem for an irreducible recurrent Markovian matrix T asserts that for all f, g e ^ 2. with g > 0 , the ratios D^(f ,g)j converge to ^ f / $ g for all j as N tends to,infinity (see [1], Corollary 1 applied to the space of non-negative integers v/ith counting measure). In discussing this theorem for matrices T which are not necessarily Markovian, one might expect that it would be necessary to assume that T is an operator. Surprisingly, this is not so. V/e give a necessary and sufficient condition that the ratios H^^(f ,g)j converge for all f, g s , g > 0 . The condition is simply that v be bounded; $0 51 i.e., that sup v- < oo . This condition is sufficient i ^ to insure that fT^ is defined for all f s for all n (althou[;h fT^ itself may not he in ), On the other hand, if v is unbounded, the expression fT^ may fail to be defined for some f e some n , in which case we cannot have a ratio theorem for all f,g e , or the expression may be defined for all f e Sind for all n . In this latter case we exhibit vectors f,g G: , g > 0 such that

lim sup D_(f,g). = CO N — >00 ° and lim inf D.y(f ,g) = 0 . K —> CO *’ ° To obtain convergence of the ratio D^^(f ,g) under the assumption that v is bounded, we may use the trans formation p^.1,J = t l.J ? . V./v. ,j X . Let f,S G ^ X , s > 0 .

Then fv , gv e gv > 0 . V'/ej have 2. f. t ? . (4.1) Ü = ^n=o -"i *^i ^ij

< : o 52 and the last expression in (4.1) converges to fv )/(^ gv ) by the Chacon-Ornstein theorem. Clearly a necessary and sufficient condition that (fT^)j be defined for all f e is that

(4.2) sup t^.. < 00 . i We note parenthetically that boundedness of v implies (4.2) for each j , n . For let M = sup v. ; then i ^

n ' =k ■^ik ’'k 2: Vj so that

and (4.2) follows, V/e now consider the case when v is unbounded; i.e,, sup v. = oo . 1 ^ Theorem 4.1. Let T be an irreducible recurrent matrix satisf^'^ing (1.?) and (4.2) for all j,n . Suppose that the right-invariant vector v is unbounded. Then there exist vectors f,g c , g > 0 such that

lira sup D.,j(f,g) = CO K — > GO " ® (4.3) lira inf D,-(f,g) = 0 . K *-> 00 °

Proof. . Let M = sup t? , K = K,. , and ^ Ü. V iX -k "V 53

S(lN,f) = . All limits in this proof are taken as N tends to infinity. Let

h = = 2'6co

Si = whore Sj^ = 5^^ .

Then

lim %(f]_,S2^o C$^iv)/(k'iv) = 2 by Theorem 2.3. Thus there is an such that

(Lor consistency set = 1 ). There exists IC^ > such that

Let Yo = where y = 6 . /2^» M... 2 "'j ■'0=0 " "j,K2 "'l ^2 " ^1

Sp = Si + Ï2 •

Then

SCN^^Yg) < 1/2'

and

lim ^];t(^2’^2^o ^ ($f2^)/($G2v) < ($f^v)/(^y^v) < 1/2 . 34 Thus there is an Ng > such that

.Ve v;rite out one more step before doing the induction. There exists > Kg such that

> %T •5^*($6pv) .

Let 3 s 00 (p^ = ((P: i.

^3 ' ^2 +

S3 " S2

Then sdig.iPg) < 1/3^ and

lim Dg(fg,g^)g - (^fjvi/C^EjV) > ($ 3 .

Thus there is an > Ng such that

Suppose now that

< Kg < ... < Kg^_;L < Ng < ... < Ng^_^

fl, fg, '"'*^2n-l ^3» ^5 ’ •••* ^2n-l

Si» 6 2 » •••» 62n-i ^2» Y^» *, Ï2n_2 55 have been determined in such a way that for m = 2,...,n ,

^2m-l “ ^2m-l + ^2m-) + *•* + +/%

^2m-2 “ ^2ra-3

^2m-l “ ^2m-2

S2m-2 = Y2m_2 + ^2m-4- + " . + Yg +

^N2j„_2^^.2m-2»®2m-2^o ^ l/(2m-2)

S(K2m-3’W 2 ) ^ l/(2m-2)2

^(''2m_2'"2m_l) ^ V(2n,-1)2 .

Then there exists > ^2n-l ^^^h that

^ ^^^2n-l (^^2n-l"^^ "

Let

^2n " ^2n-l

^2n “ ^2n-l ^2n Then

S("2n-l'Y2n) ^ l/(2n)2 and lim Dj,^^(f2^,g2„)„ < <$f2n_x'')/dY2„v) < l/2n . 56 Thus there is an such that

There exists such that

%n+l " %„ • Let

“’2n+l “

^ 2n+l “ ^2n + ^2n+l

^2n+l = ®2n Then

®(''2n-

lin ®K^^2n+l’®2n+l^o - > 2n + 1 .

Thus there is an ^ ^2n that

%2n+1^^2^+^'®2n+l^o >2^ + 1 •

In this way sequences of vectors cp^, cp^, .,, and Yg) Yz}.» ••• are determined. VJe set

f *® f2 + cp^ + (p^ + ... 37 and

S = Si + Ï2 + Yzj. + ••• •

Now ||cpj^|| < k”^ and ||Yy| < k”^ for all k since

= 1 ; hence f » g e . Clearly g > 0 . Also

> ^^^2n-l*^2n-l) ^(^2n-l* ^2n + ^2n+2 + ' “

while

^(^2n'^2n^ ^ ^^^2n^ ^2n+l + ^2n+3 + "') ^ S(N2^,S2j^)

!^^2n:£2nllJ: ^(^2n'G2n) “

% 2n^^2n»S2n^ + ^/G(^2n*S2n^ ^ (n— >oo) .

Thus we have constructed f and g such that (4.5) holds. V. THE STRONG RATIO LIMIT PROPERTY

An irreducible aperiodic matrix T satisfying (1*7) is said to have the strong ratio limit property (SRLP) if there exist positive numbers and , i = 0, 1, ... such that

(5.1) ^ ■'iVVh for all m = 0,+l,+2, .... It can be proved that the assumption of aperiodicity implies that for fixed i and j t t ? . > 0 for all n sufficiently large; thus the above ratios make sense for all n sufficiently large. In the recurrent aperiodic Markovian case Orey [73 proved that SRLP is equivalent with the convergence of to 1 , where u^ is the n-step transition probability from a fixed state to itself. Here we show that Orey*s result, and method of proof, extend to matrices T . We also consider the periodic case, which requires only minor modifications in proofs. As noted in the introduction, the assumption that T satisfy (1.7) is equivalent with the assumption that r = 1 , where r is the (common) radius of convergence of the series t?^ s^ . The study of the general case r > 0 reduces to the study of the case r = 1 . The case r = 0 is considered in Proposition 5.1 and the subsequent example.

38 39

We state some additional definitions and results needed in this chapter. For k > 0 we define

(3.2)

One easily shows that for all i»j,k > 0 and for all m,n > 1

( 5 . 3 ) •

The following ^renewal^^ equations (see [11] ) are essential to our work;

and

(5.5) + o ^ h •

We include the proof of (5.5) for the sake of completeness. The proof of ($.4) is similar. For n = 2 we have

^ij “ ^ i ^ o ^ u e ^^0 ' ^io “

oîo ôj oîj •

Assuming that (5.5) holds for n , we consider t^î :

- “ = ^ o ^ : i o < o +

"=a“ o o^iA o*L ^oi^ + o^?f * 40

o"îo ôj “ ^k=l ^ o"îo ôj “ 4-k . n+l-k ^ .n+1 ^k=l oîo ôj + oîj •

This completes the proof of (5«5)- If T is irreducible, we can see from (5.4) and (5.5) that for given i and j there exist integers m and p such that

(5.6) > 0 and > 0 .

We will use these facts later. We set

(5.7) and .

Then equation (5.4), with j = 0 , reads

(5.8) ffc "n_k •

Finally, for 0 < s < 1 , v/e define

Gij(s) = "îj k"îj(^^ “ ^r^l k"îj '

For 0 < s < 1 v;e have

(5.9) G. . (s) = 1--- — . 1 - i^^ii^s) The proof for Markovian matrices (e.g., in [5], p. 45) carries over without change. If T is recurrent, (5.7) and (5 .9 ) imply

(5.10) fn - : -

Pronositlon 5.1. If T is an irreducible aperiodic matrix with r = 0 , then 41

(5.11) lim sup(u ,/u ) = OD . n — > CD Proof. If lim sup(u -/u ) < 0 0 , then there exist n —> 00 ^ numbers M and Nsuch that u^^^/u^ < M for n > N , Then u^^^ < for n = 0, 1 , ... . Hence lim sup I u I < M and r > > 0 $ which is impossible, n — > 00 The following example shows that lira sup in (5 .11) cannot be replaced by lim • Define sequences f_XI and u ri ( ^ by •^2n—1 ” ^ * ^2n ™ ^ for n*=l, 2, ••• ; u^ = 1 ,

"n " ^k=l %_k • "2n-l Z fgn-l *hat the

series u^ has radius of convergence zero. One

shows by induction that ^2n-l - ^*^2n-l * ^rom which it follows that

^k=2 ^2n+2-k - ^^”^^^2n-l ^2n-l -

2(n-l)(fg^_^)^ < f2n+l '

Applying this to

^2n+2 ^ ^1 ^2n+l 1 ^2n . ^2n+l ^1 ^2n+l ^2n+l ^2n+l ^ 2n+2-k ^2n+l

one obtains that ^2n+2'^^2n+l £ 1 “ 2* 2^ + 1 ;

hence lira C^2n+2^^2n+l^ < w • For i = 1, 2, ... , 4-2

^2i,o “ ^ » ^21-1,21 “ ^2i+l'^^2i-l »

^00 “ ^ol ~ ^1 * Sind j = 0 otherwise, to obtain an irreducible aperiodic matrix T = (t^^ . ) for which o'^oo = *So = “n We nov; state our main result. Theorem 5.2. If T is an irreducible recurrent aperiodic matrix satisfying (1,7), then T has SRLP if and only if

(5 .1 2 ) ^n+l/^n — ^ ^ (n— $>oo) .

The vectors tc = (ti^, tu^,...) and r = (t ^, .... ) are left (resp.) right-invariant under T . Proof. The only if part is obvious. To prove the if part, we rely on the following lemmas which are stated under the same hypotheses as Theorem 5*2.

Lemma 5.5. ^n^^n — ^ ^ (ii— > oo) .

Proof. By (5.8) we have

^n " ^k=l %-k " ^k=l ^k ^n-k + ^n ' This implies that

^n/^n = 1 “ ^k=l ^k ^n-k^^n * By Patou’s lemma, (5.12) and (5.10),

n — y 00 43 hence lim sup(f /u ) < 0 . The result follows since n — > 00 ^ ^ V “n ^ ° - Lemma 5.4. For given i and j , there exist numbers c and 5 independent of n , such that

^ <= • Proof. If i s= 0 = 0 , the lemma is obvious. Other wise, by (5 *6 ) there exist integers m and p such that o^oi ^ ^ o^Jo ^ ^ • ^tien, using (5*5) i we have

o ^ o T ’" ^ 0

^ o^oi i ^ 0 . d - 0

2: o^?d o*do i " 0 . d ^ 0 . from which the lemma easily follows.

Lemma S.S. Por any i and j , o^ij'^^oo — ^ ^ (n — > oo) . Proof. By Lemma 5*4, there exist numbers c and s such that

(5.15) -o^?d ^ , < = _o * ^ : r C = ^n+s •— "n.s • OO 00 00 n+s n

By Lemma 5*5 and (5.12), the right side of (5*13) tends to zero as n tends to infinity. 44

To prove the theorem we firstconsider the ratio t^^/t^o • n , N with 0 < N < n we let

^n-k .k .n-k %.n = 4.1 2^4-^ • 00 oo Then by (5.4), we have that t^j/t^^ “ "^N n ®N n * will show that

(5,14) lira (lira sup ) = 0 , N— n-»oo from v/hich it follows that, given e > 0 ,

for all N sufficiently large, and hence that .k lim^sup(t“ ,/t^o) < .t

By Fatou’s lemma

n — > 00 so that (5.14) implies the convergence of t^j/t^^ to _t^. . (That this limit is finite follows from IV —JL O OJ

Lemma 5.4- and (5.10); that it is positive follows from (5.6)), Now to show (5*14), By Lemma 5.4-, there exist numbers c and s such that _t^. < c • Hence, O OJ — O 00 using (5 .7 )t we have that 45

^k+s ^n-k T> ^ m. O 00 00 _ ^ ^k+s ^n-k %,n - ^ ^k=N+l 7n “ ® ^k=N+l un 00 Changing the index of summation and writing this sum in two parts * we have

T^n.+s ^ k % + s - k ^N+s ^ k ^ n + s - k cZn ^k+s ^n-k k=N+l ^ k = l ÛZ ------^ k = l --- u ------u,n n n which, by (5 .8 ), equals

u.n+s ^N+s ^k ^n+s-k - ^k=l Ûn---- and this, by (5 .1 2 ), approaches c(l - as n 00 . Finally, by (5.10), c(l - f^^) — > 0 as

N 00 ; hence (5«14). We define

" ^iSl o^oj • Next we consider the ratio t^./t^ to which we ij oo apply a similar argument. We set

. k .n-k ^n-1 o io 0,1 =N.n ° 4 . 1 % . n ^k=N+l “ .n oo 'oo and show that lim (lim sup D., „) = 0 , Applying (5.5) N — > 00 n —^ 00 * and Lemma 5«5t one obtains that ^k^l o^io 46 which is finite and positive. Again by Lemma 5,4 there exist numbers c and s such that _t9_ < _t^*^ , so O lO ~ O 00 that ^k+s .n-k _ £ 2 2 1 DN,n » - < O s ---.n 'oo

^k .n+s-k .k ^n+s-k Q -n+s-1 0^00 ^0.1 ^N+s 0^00 ^o.i nc=i “k =1 4_n oo

4.n+s +n+8 4.n+s-k 0 ° _ -N+s ^k ^k=l tn _ t^ 00 t^00 oo which approaches «N+S (5.15) C It . (1 u - =k.i 4 >

as n --> CO When N — 9» 00 « (5 .1 5 ) tends (5.10). We set

Ti ■ \ “= i and observe that . n+m 4-n+m 4.n+m oo “ % ^kh oo ' C "

As n — > 00 t the middle tvrm on the right converges to 1 while the first and last terms converge to and i 0 47

respectively; this proves that (5 *1 ) holds. To prove invariance of u , we observe that

îô î îj “ ^:So^k^l oôi îj “ îSlî/o oôi îj +

^k^l 0^00 ôj “ îSl oôj^ + "ôj “ îSl 0^0j " • A similar argument establishes invariance of v . Corollary 5.6. If T is an irreducible recurrent aperiodic matrix satisfying (1.7)» and if T has SRLP, then the limit in (5 *1 ) is independent of i and k if and only if T is Markovian. Proof. The limit in (5*1) is independent of i and k if and only if the vector v is constant, say Vj^ = 1 for all i . But since v is right-invariant, v^^ = 1 for all i implies that T is Markovian. Conversely, if T is Markovian, it is well-known from the theory of

Markov chains that o^io ” ^ for all i , and hence V is constant. We mention the following theorem, which was proved by Orey [73 and Pruitt [83 in the Markovian case. Pruitt’s proof is purely analytic and goes through in our situation. Theorem 5.7. Let T be an irreducible recurrent aperiodic matrix satisfying (1.7). If there is an integer d such that 48

lim sup < 1 , n — > CD ^nd then (5 .1 2 ) holds. The Periodic Case. If the matrix' T is irreducible and periodic; i.e., ■with period d > 1 , we cannot have SRLP in the form of (5«1) because t?. is positive for only certain values of n , which are interspersed among the integers in multiplies of d units. Given an irre ducible Markovian matrix P = (p..) with period d , for each i and j there is a unique integer 6 = 6 (i,j) , 0 < 6 < d , such that for all k sufficiently large, pkd+6 (i,j) > Q ^ The proof, e.g., in Kemeny and Snell

[4], pp. 5-7, does not depend on the Markovian (or finite) character of P ; hence the result holds for an irreducible matrix T with period d . Such a matrix is said to have SRLP if there exist positive numbers and , 1 = 0 , 1 , ... such that .(n+m)d+6 (i,j)

^kh ^ ^ for all m = 0,+l, ... . (For d = 1 , (5.16) coincides with (5 .1 )). We have 49

Theorem Let T be an irreducible recurrent matrix satisfying (1.7), and suppose that T has period d . Then T has SRLP if and only if

(5.17) "(n+Dd'^'^nd ^ (n— » m ) .

The vectors n ~ (ti^, ...) and t = (t^, ...) are left (resp.) right-invariant under T . A slight modification in the proof of Theorem 5*2 yields a proof of Theorem 5.2^. We state the lemmas in their modified form and omit the details. In Lemmas 5.4^ and 5 .5 ' , e = 6 (i,j) .

Lemma 5.%'. ^nd^%d — ^ ^ —» cd) .

Lemma 5.4^. For given i and j , there exist num bers c and s , independent of n , such that <

j_(n+s)d ^ O'OO .nd+e /*.nd Lemma 5.5 . For any i and j , o^i^j '^^00 — ^ ^

(n — > 00) . Corollary $.6 remains true if the word aperiodic is deleted and (5»l) is replaced by (5 .16). Let (q..)IJ be the minor of T^ composed of the rows of index h and columns of index k for which 6 (0 ,h) = 6 (0 ,k) = 0 . Then (q..) is an irreducible recurrent aperiodic matrix 50

satisfying (1.7). Applying Theorem 5-7 to (q..) one obtains Theorem 5.7^ * Let T be an irreducible recurrent matrix satisfying-- (1.7), and suppose that T has period d . If there is an integer e such that

lim sup < 1 , n — > CD n e then (5 .1 7 ) holds • REFERENCES

1. Chacon, R. V., Identification of the limit; o f operator averages, J, Math. Mech., 11 ( 1 9 & 2 ) , 961-968. 2. Dunford, N. and J. T. Schwartz, Linear Operators I, Interscience Publ., New York, 1958. 5. Karlin, S., A First Coarse in Stochastic Processes, Academic Press, New York, 1966. 4-. Kemeny, J. G. and J. L. Snell, Finite Marieov Chains, D. Van Nostrand Co., Inc., Princeton, N. J • , 1965. 5. Lorentz, G. G., A contribution to the t h e o r y of divergent sequences, Acta Math. , 80 (194-8) ^ 167-190. 6 . Orey, S. , Anergodic theorem for Markov cliains, Z. V/ahrscheinlichkeitstheorie, 1 (1962), 7^-176. 7. _____ Strong ratio limit property. Bull. A m e r . Math, SCO., 67 (1961), 571-574. 8 . Pruitt, W. E,, Strong ratio limit property for r-recurrent Markov chains, Proc. Amer. Mat;hi, Soc., 16 (1 9 6 5 ), 19 6 -2 0 0 . 9. Sucheston, L., Banach limits. Amer. Math- Monthly, 74 (1967), 508-511. 10. _____ On the ergodic theorem for positive operators I, Z. V/ahrscheinlichkeitstheorie, 8 (1967), ZL— 11. 11. Vere-tjones, D., Geometric ergodicity in d_enumerable Markov chains. Quart. J. Math. Oxford (2 5, 15 (1962), 7-28.