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Pacific Journal of Mathematics GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM S HU -T EH C HEN M OY Vol. 11, No. 2 December 1961 GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM SHU-TEH C. MOY 1«, Introduction, Let X b e a non-empty set and ^ be a <7-algebra of subsets of X. Consider the infinite product space Q — n»=-~ Xn where Xn — X for n = 0, ± 1 , ±2, • • • and the infinite product tf-algebra J^~ = II~=-oo ^ where ^—S/1 for w = 0, ± 1 , ± 2 , • • • . Elements of -0 are bilateral infinite sequences {• • •, X-19 x09 x19 • • •} with xn £ X. Let us denote the elements of Q by o). If <w = {•••, #_!, #0, #2, • • •} xw is called the nth coordinate of co and shall be considered as a function on Q to X. Let T be the shift transformation on Q to 42: the nth coordinate of Too is equal to the n + lth coordinate of co. For any function g on Q, Tg is the function defined by Tg(a)) = 0(Ta>) so that T&w = jcn+1. We shall consider two probability measures ft, v defined on JC Let Qn = n*=i X* where X4 = X, i = 1, 2, • • •, n and ^ = n?=i ^ f where ^f=s^i = 1,2,---,n. Then 42X = Z and ^ f = ^ Let ^ , n , m ^ w , w = 0, ± 1 , ± 2 , •••, be the (7-algebra of subsets of Q consisting of sets of the form [a) = {•••, #_!, o?o, o;3, • • • } : (xm, xm+1, •••, x j e E] where E e ^ L w 4 1 . Let ^oo lW be the a-algebra generated by Um=-i ^ . » Let j«m>w, vm<n be the contractions of p, v, respectively, to J^,n and P-^^, v-co,n be the contractions of p, v, respectively, to J^i^^. Throughout this paper vm>n is assumed to be absolutely continuous with respect to Pm,n, vm,n < Pm,n, for m < n, n = 0, ± 1 , ±2, • • •. Let/ m ,« be the derivative of vm>n with respect to ftm,n9fm,n = dvmjdpm>n. fmM is ^ n measurable and nonnegative. / m w is also positive with v probability one. Hence Vfm.n is well defined with v probability one. A fundamental theorem of Information Theory by Shannon and McMillan may be considered as a theorem concerning the asymptotic properties of fm>n as n—* oo. The theorem may be stated as follows: Let X be a finite set of K points and Sf7 be the a-algebra of all subsets of X. Let v be any stationary (T invariant) probability measure on j^~~ and p. be the equally distributed independent (product) measure. Then wUog/i,,, converges in L^v). In particular, if v is ergodic, the limit function is equal to log K — H with v probability one where iTis the entropy of v measure [3] [8]. Generalizations to arbitrary X, c/7 were first studied by A. Perez. He introduced an A,,, condition on v as follows, v is said to satisfy A^ condition if v_oo,» is absolutely continuous with respect to v_OOtQ,p1>n for n = 1, 2, • • •. He proved the following theorem. If vf p are stationary and p is the product (independent) measure on ^ and if Received May 31, 1960. This work was supported in part by a summer research grant from Wayne State University. 705 706 SHU-TEH C. MOY (a) limn_>oo n"M log f1>ndv exists and is finite, (b) v satisfies condition AM then {n~l\ogf1>n} converges in L^v) [6]. Later Perez announced that the theorem remains to be true for any stationary measures fi, v [8]. The present writer proved that for Markovian fi, v with v being stationary and fi having stationary transition probabilities the v-integrability of log/i,2 implies the Lx{v) convergence of {nr1 log fun}. The proof is based on an iteration formula for f1>n [4]. In this paper we shall study the case that v is stationary and fi is Markovian with stationary transition probabilities. It shall be proved that the condition (c) j (log/1)W - log f^-Jdv g M < co for n = 1, 2, 3, • • • implies the L^v) convergence of {wMog/jJ. In fact the conditions (c) and (a) are equivalent for this case, so that the theorem is a generalization of the theorem of Perez given in [6]. The proof is conducted along similar lines used by McMillan. The crucial step is proving the L^v) convergence of {log/_w0 — log/-w,-i}. The condition (c) is shown to be necessary and sufficient for this convegence. 2» Generalizations of Shannon-McMillan theorem* Let x, £? Q, J^i2nf^nf^.n,t*m.nfVm.n?fm.n be as in /. Notations for conditional probabilities and conditional expectations relative to one or several random variables will be as in [2], Chapter 1, § 7. A probability measure on ^ is Markovian if, for any A e t 5 f m < w w = 0, ± 1 , ±2, • • • P[xn e A I xm, • • •, xn^] = P[xn 6 A I a?n_J with probability one. A Markovian measure is said to have transition probabilities if for any A e y and any integer n stationary P[xn e A I Xn-J = TnP[x0 e A \ a.J with probability one. In this paper, since we have two probability measures /^, v, we need to use subscripts ft, v to indicate conditional probabilities and conditional expectations taken under ft, v respectively. For any E c Q,IE, the indicator of E, is the real valued function on Q defined by IE(o)) =z\ if a) e E = 0 if co $ E . The log in this paper is the logarithm with base 2. LEMMA 1. (1) Define » v'm,n on J*£ffI by GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM 707 then v'm>n is a probability measure on ^ > w with v'Wtn(E) — vmin(E) for E e J ^ - x . Furthermore vm<n < v'm<n with Proof. I ^ . n - ! I Xm, • • • , X = \ fm.n-idp . JE Hence v'mM is a probability measure on ^ , w . Furthermore, for E e ^ , , fm.ndfi - [ (fm.Jfn.n-l)fm.n-ldf* JE JE Hence vm>n is absolutely continuous with respect to v'm>n and dvm>nldv'mtn — Jm.nlJm.n—i* THEOREM 1. If v is stationary and fx is Markovian with stationary transition probabilities then \ ** ) Jm,nlJm,n—i •*• \Jm—n,oUm—n,—i) with v probability one for all m < n, n — 0, ± 1 , ± 2 , Proof. If fi is Markovian and has stationary transition probabilities then for any A e Sf Py\.®n ^ A I Xm, • • •, #w-iJ = r\\Xn 6 A %n-\\ n - T P»[x0 e A I aj.J with ix probability one and, therefore, also with v probability one. Hence for any Ae^fBe ^T_m v'm,n[xn e A, (xm, . ••,&„_!) e 5 ] = \ Pt&n 6 A I flJTO, • 708 SHU-TEH C. MOY TnP,Hx0 e A = [ J[(xm_1,---,a;_1)6£] xQ e A I x ^ It follows that *4,»[(a«, • • •, o?n) e C]i= v'm-n>o[(xm-n, • •., a;0) e C] for every C e ^ _ m + 1 . ^^m.nl^^m.n Since by Lemma 1 ~ Jm,nlJm,n-l> ^^m-n.ol^^m-n.O == Jm—n,olJm-n,—l (2) follows easily. LEMMA 2. 7/ /^ is Markovain extension of i42,0 ^ Proo/. and m1 < m2 < 0 t/ieti ^ l t 0 is an For any A e £? /3 e ^T m 2 v'mv0[x0 e A , (ajTOa, • • •, tf-x) e B ] *[%o e A \ x m , ---tX^ = \ = ^m2lo[^o e A,(a?TO2, • • -, a?_i) e B] . It follows that for every E e THEOREM 2 2. If fi is Markovian and m1 < m2 < 0 (3) Proof. By Lemma 2 i4 li0 is an extension of v«it0 to ^ 1 § 0 . Since ^ . o < ^ . o , ^ 2 ( 0 < v;2,0 by Lemma 1, dvmjdv'm2>, is the conditional expectation of dvmitOldv'mvO relative to ^ 2 , 0 under the measure v'mim0. Jensen's GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM 709 inequality for conditional expectation implies that 0 Hence (4) 0 g J log (dvmjdv'mjdv ^ j log (dvmJdvmv0)dv and (3) follows from (4) and Lemma 1. THEOREM 3. / / [i is Markovian then {log/wl 0 — log/TO,-i} converges with v probability one as m —> — oo. The limit function may take ± GO as its values. Proof. It is sufficient to prove that {/m,-i//wM} converges with v probability one as m —> — oo. Since vmi0 is absolutely continuous with respect to v'm>0 and dvmiQldv'm>0 = fmjfm,-i by Lemma 1, fm,-Jfm.o is t h e derivative of vmi0 continuous p a r t of v'm>0 with respect to vm>Q. Since, by Lemma 2, ^ 1>O is an extension of i4 lp0 if mx < m 2 , {—/-fc)-i//-fc>0, ^*.o> fc ^ 1} is a v semimartingale ([2] p p . 632). Since the semimartingale convergence theorem implies that {f-k-Jf-kiQ} converges with v probability one as k —> oo. The following lemma may be considered as an improvement of a theorem by A. Perez ([6] Theorem 7; pp. 194). LEMMA 3. Let /31 c j32 c • • • 6e a sequence of a-algebras of subsets of Q and /3 be the a-algebra generated by \Jk /3k. Let </>, X be two probability measures defined on ft and 0ft, Xfc be the contractions of </>, X, respectively, to ftk. If $k is absolutely continuous with respect to Xk for k = 1, 2, • • • and if there is a finite number M such that \ log {d$ for k — 1, 2, • • ( i ) cj> is absolutely continuous (ii) log (d<PldX) is <fi integrable ^ M with respect to X, and there exists Km I log (d<t>kldxk)d4> = \ log {d<t>\dX)d<f> , 710 SHU-TEH C. MOY (iii) {log (d4>kld\k)} converges in Lx(4>) to log (d(f>ldX). Proof. ( i ) Let hk — d</)JdXk. Then {hk, /?*., k ^ 1} is a martingale under X measure. Now M ^ j log (d<t>kldXk)d<t> = [ (log hk)hkdX . and ( 5) M + i ^ [(/^ log ft* + J ) d \ ^ (log ^) ( ftfcdA, } Hence ( KdX^ilognyW+i) J(hk^n) so that I hkdX —> 0 as % —• oo, uniformly in k. Hence {hk} converges with X probability one and also in LX(X) ([2] Theorem 4.1, pp. 319). Let the limit function be h. Then I hdX = 4>(A) for all A e \Jkftk and so for all A e /3. This proves that <£ is absolutely continuous and that h = (d^/dX). (ii) The sequence {/^ log Z^} converges with A, probability one to h log /&. Since the functions hk log /^ are bounded below uniformly by the number J, \h log MX ^ lim lfcfc log /ifcd\ — lim^l log hkd<t> ^ M . Hence hlogh is X, integrable. Since the real valued function | l o g | is continuous and convex, h± log /^, h2 log /t2, • • •, h log ft constitute a semimartingale under the measure X([2], Theorem 1.1, pp. 295). Hence \h± log hxdX ^ \ft2 log ft2dX ^ • • • ^ ft logftdA,, so that lim^oolftfclogftfcdX exists and is equal to \ftlogftdX. Now I I log h\d<t> — \ft I log ft I dX — I IftlogftI dX, , hence log ft is ^ integrable and ( 6) (logftd<£= Ift logftcfX,= lim \hk log ftfc dX = lim \ log ftfc d0 . J 1 J k->°o J Inequality (5) was pointed out by the referee. shortened by following his suggestions. fc-*oo J The proof of Lemma 3 was much GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM 711 (iii) Since hx log huh2\ogh2T • • •, h log h constitute a semimartingale under the measure X, we have, for E e /3kf l hjcloghjcdX g \ hk+1 loghk+1dX g \ JE JE hloghdX. JE Hence \ loghkd(f)^ \ loghjc^dcj) ^ \ \oghdcj> , JE JE JE so that log 7^, \ogh2, • • •, logh constitute a semimartingale under the measure 0. Hence (ii) implies that log hk are uniformly <f> integrable and {loghjc} converges to log h in L^) ([2], Theorem 4.1s, pp. 324). THEOREM 4. If ft is Markvian and there is a finite number M such that ^,-^dv rg M for m— — 1, —2, ••• then {log/m>0 — \ogfm<_^\ converges in Lx(v) as m —* — oo. Proof. By Lemma 2 v'mv0 is an extension of vrm^ if m1 < m2 < 0 and If there is a probability measure i/ defined on the tf-algebra generated by Um=-i^m,o which is an extension of v'm>0 for m = — 1 , —2, •••, then the conclusion of the theorem follows easily from Lemma 3. If X is the real line and if S^ is the tf-algebra of Borel sets then the existence of i/ follows from the Consistency Theorem of Kolmogorov. For the general case we shall proceed by using the usual representation by space Qf of sequences of real numbers as follows: Let 9lC — / ~JC,0lJ -16,-1 * L e t G b e t h e m a p of Q into t h e space Qf of real sequences { £ , | 2 , • • • } defined b y G((o) = Considering £k as functions on Qr we have - gk(co) . Let /3k be the collection of Borel subsets of Q' which are determined by conditions on £lf | 2 , • • •, gfc and ft be the collection of all Borel subsets 712 SHU-TEH C. MOY of Q'. Let 0 be the probability measure on /3 and </>fc, Xk be the probability measures on j3k defined by Xk(E) = v'-UG'lE) . {gk} converges in L^v) if and only if {£&} converges in L^). Now Xk are consistent; Kolmogorov's Consistency Theorem implies the existence of a probability measure X on ft which is an extension of every Xk and d(f>kldXk — ffc. Hence Lemma 3 is applicable and the Z/i(</>) convergence of {£,.} is obtained. THEOREM ary transition 5. If v is stationary probabilities and if and pt is Markovian with station- and if there is a finite number M such that for n — 1,2, ••• then n~l\ogfQ%n converges in Lx(v) as n-+oo. In particular, if v is ergodic, the limit is equal to a nonnegative constant with v probability one. Proof. By Theorem 4 {log/ m0 — log/^.-J converges in Lx(v) as Let h be the Lx{v) limit of the sequence. Let h be the m_>_oo. Lx{v) limit of the sequence {n'1 £?=i T'h}. By Theorem 1 /0>B//0,n-i = r n (/- B . 0 //- w ,-i), hence -1 logr/o.. = n-1 log/ 0 , 0 + w1 £ 2" log (f-Jf-t.-i) ^ W-1 t j I T* log (f-i.Jf-i.-r) ~ T'h I dv ^~x S r*^ - h I dv -> 0 as n -> oo . Thus the Lx(v) convergence of {^"Mog/o.J is proved. The limit is h GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM 713 which is the Lr{v) limit of {n'1 2?=i T*h}. If v is ergodic h — \hdv with v probability one and \hdv - lim [[log/w,o - log fn.-Jdv ^ 0. COROLLARY 1. Under the hypothesis of Theorem 5 if v is and ergodic but not Markovian then v is singular to pt. stationary Proof. If [X is Markovian but v is not Markovian then there is a positive integer n0 such that i"[/o.» o - 1 =3 fc /o.n o ]>O. For, if for every positive integer n fAAn-i * An] = 0 then jry\Xn 6 J± XQ, • • •, # w _ij — ±y\Xn 6 A. \ ^ w -iJ w i t h v probability one for every A e y N o w since Jo,no-i = a n d v is M a r k o v i a n i n s t e a d . -^^L/o.Wo I ^o> * * * 9 %no-i\ and the function I;log! is strictly convex, hence 5/o.»0los/o,«0d/i - JZo.no-! log/o,»o-idj" > 0 so that 5[logZo,,0 - logZo.»0-i]d^ > 0 . Since \[logZo,» — logZo,w-i]^^ is non-decreasing in n, lim l[logZo,» - logZo>w-i]^^ = a> 0 . Now v is ergodic; the Lx(v) limit h of {^"MogZo.J is equal to a with y probability one. Let nlf n2, • • • be a sequence of positive integers for which {wjb"1 log Zen*.} converges with v probability one to a so that {1/Zo.nJ converges to 0 as nk—> oo. Let ^ ' be the ^-algebra generated by \Jn^o,n and let fi^,, [i^, be the contractions of [i, v, respectively, to _^ r '. Since 1/Zo,n is the derivative of v-continuous part of fJt0i7l with respect 714 SHU-TEH C. MOY to vOin, {l//o.»} converges with v probability one to the derivative of v-continuous part of fi' with respect to 1/ by a theorem of Anderson and Jessen [1], Now we have lim l// 1>n - 0 with v probability one and fi is singular to i/. Hence fi, v are singular to each other. Extensions of Theorem 5 and Corollary 1 to if-Markovian fi are immediate. 3. Discussion* As was mentioned in the introduction the crucial step in establishing Theorem 5 is to prove the L2(v) convergence of {log/_W(0 — log/- w ,-i}. If fi is the product (independent) measure on ^ the measure vf in the proof of Theorem 4 is actually v_oo-i x fiQ>Q. Thus condition (c) or, equivalently, condition (a) implies condition (b) in the introduction. In [7] it is stated that the condition (b) is necessary for the LXv) convergence of {log/_„,„ —log/_„,_!} ([7] Theorem 2(b)). A simple is as follows. Let X be the real line and £f be the collection of all Borel sets. Let v — fi and distribution of x0 be Gaussian. Let v(x0 = xx) = fi(x0 = xx) = 1. Then v_li0 is singular to v_1(_! x v0>0, however the Lx(v) convergence of {log/_ wo — logZ-^.J is trivially true since f = 1 REFERENCES 1. Erik Sparre Anderson and Borge Jessen, Some limit theorems in an abstract set, Danske Vic. Selsk. Nat.-ys. Medd. 22, No. 14 (1946). 2. J. L. Doob, Stochastic Processes, John Wiley and Sons, Inc., New York. 3. B. McMillan, The basic theorems of information theory, Annals of Math. Statistics, 2 4 (1953), 196-219. 4. Shu-Teh C. Moy, Asymptotic properties of derivatives of stationary measures, Pacific J. Math., 1O (1960), 1371-1383. 5. A. Perez, Sur la theorie de V in formation dans le cas d'un alphabet abstrait, Transactions of the First Prague Conference on information theory, statistical decision functions, random processes, Publishing House of the Czechoslavak Academy of Sciences, Prague, (1957) 183208. 6. , Sur la convergence des incertitudes, entropies et informations echantillon (sample) vers leurs vraies, Transactions of the First Prague Conference on information theory, statistical decisions functions, random processes, Publishing House of Czechoslavak Academy of Sciences, Prague (1957), 209-243. 7 , Information theory with an abstract alphabet, generalized forms of McMillan's limit theorem for the case of discrete and continuous times, Theory of Probability and its Applications, (An English translation of the Soviet Journal Teoriya Veryoatnostei i ee Primeneniya) Vol. 4, No. 1 (1959), 99-102. 8. C. E. Shannon, The mathematical theory of communication, Bell System Technical Journal, 2 7 (1948), 379-423, 623-656. SYRACUSE UNIVERSITY PACIFIC JOURNAL OF MATHEMATICS EDITORS RALPH S. PHILLIPS A. Stanford University Stanford, California University of Southern California Los Angeles 7, California F. H. BROWNELL L. L. J. University of Washington Seattle 5, Washington WHITEMAN PAIGE University of California Los Angeles 24, California ASSOCIATE EDITORS E. F. BECKENBACH T. M. CHERRY D. DERRY M. OHTSUKA H. L. ROYDEN E. SPANIER E. G. STRAUS F. 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Paige at the University of California, Los Angeles 24, California. 50 reprints per author of each article are furnished free of charge; additional copies may be obtained at cost in multiples of 50. The Pacific Journal of Mathematics is published quarterly, in March, June, September, and December. The price per volume (4 numbers) is $12.00; single issues, $3.50. Back numbers are available. Special price to individual faculty members of supporting institutions and to individual members of the American Mathematical Society: $4.00 per volume; single issues, $1.25. Subscriptions, orders for back numbers, and changes of address should be sent to Pacific Journal of Mathematics, 103 Highland Boulevard, Berkeley 8, California. Printed at Kokusai Bunken Insatsusha (International Academic Printing Co., Ltd.), No. 6, 2-chome, Fujimi-cho, Chiyoda-ku, Tokyo, Japan. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATION The Supporting Institutions listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its content or policies. Pacific Journal of Mathematics Vol. 11, No. 2 December, 1961 Tsuyoshi Andô, Convergent sequences of finitely additive measures . . . . . . . . Richard Arens, The analytic-functional calculus in commutative topological algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michel L. Balinski, On the graph structure of convex polyhedra in n-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. H. Bing, Tame Cantor sets in E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cecil Edmund Burgess, Collections and sequences of continua in the plane. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. H. Case, Another 1-dimensional homogeneous continuum which contains an arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lester Eli Dubins, On plane curves with curvature . . . . . . . . . . . . . . . . . . . . . . . A. M. Duguid, Feasible flows and possible connections . . . . . . . . . . . . . . . . . . . Lincoln Kearney Durst, Exceptional real Lucas sequences . . . . . . . . . . . . . . . . Gertrude I. Heller, On certain non-linear opeartors and partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calvin Virgil Holmes, Automorphisms of monomial groups . . . . . . . . . . . . . . . Wu-Chung Hsiang and Wu-Yi Hsiang, Those abelian groups characterized by their completely decomposable subgroups of finite rank . . . . . . . . . . . Bert Hubbard, Bounds for eigenvalues of the free and fixed membrane by finite difference methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. H. Hyers, Transformations with bounded mth differences . . . . . . . . . . . . . . . Richard Eugene Isaac, Some generalizations of Doeblin’s decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Rolfe Isbell, Uniform neighborhood retracts . . . . . . . . . . . . . . . . . . . . . . . . Jack Carl Kiefer, On large deviations of the empiric D. F. of vector chance variables and a law of the iterated logarithm . . . . . . . . . . . . . . . . . . . . . . . . Marvin Isadore Knopp, Construction of a class of modular functions and forms. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gunter Lumer and R. S. Phillips, Dissipative operators in a Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nathaniel F. G. Martin, Lebesgue density as a set function . . . . . . . . . . . . . . . . Shu-Teh Chen Moy, Generalizations of Shannon-McMillan theorem . . . . . . . Lucien W. Neustadt, The moment problem and weak convergence in L 2 . . . . Kenneth Allen Ross, The structure of certain measure algebras . . . . . . . . . . . . James F. Smith and P. P. Saworotnow, On some classes of scalar-product algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dale E. Varberg, On equivalence of Gaussian measures . . . . . . . . . . . . . . . . . . . Avrum Israel Weinzweig, The fundamental group of a union of spaces . . . . . 395 405 431 435 447 455 471 483 489 495 531 547 559 591 603 609 649 661 679 699 705 715 723 739 751 763