P. Billingsley. Ergodic Theory and Information. John Wiley & Sons, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....With E i (x) denoting the empirical measure, let E : fx 2 X : E n g. Then we have for each 2 M(X; with f d 6= 0 that In order to prepare the proof we recall the following well known concepts and results which will be crucial. Mass distribution principle) Caj81, Bil65] For 2 M(X) non atomic and m 2 M(X) we have for any K X that m ess sup dim(K) sup (Information cocycle) For 1 ; 2 2 M(X; such that fC 2 Z n : 2 (C) 0g fC 2 Z n : 1 (C) 0g for all n 2 N , we de ne ( 1 j 2 ) 1 (C) log 2 ( C) and h( 1 ....

P. Billingsley. Ergodic Theory and Information. John Wiley & Sons, New York, London, Sydney, Wiley Series in Probability and Mathematics edition, 1965.

....V l . The Shannon mean entropy h l of this process can be estimated by h l . The first inequality is a consequence of (3.1) P l is a stationary, but not necessarily ergodic process. We apply the corresponding version of the classical Shannon McMillan Breiman theorem (cf. 4] [1]) to this process and obtain that there is a set of trajectories l of measure one such that for each (v i ) i#Z V # l the limit (individual mean entropy) h( v i ) i#Z ) lim n## i=1 ) exists, and we have Eh( v i ) i#Z ) h l . V # l be the subset of those trajectories, for ....

P. Billingsley, Ergodic Theory and Information, John Wiley & Sons (1965)

....dimension plays in fractal geometry, where it turned out to be a suitable concept for distinguishing fractal sets from sets of a rather smooth geometrical nature. In the context of computability, originating from applications of Hausdorff dimension in information theory (see for example [4]) a close connection between Hausdorff dimension and Kolmogorov complexity was established [6, 16, 17, 18, 20, 21] and the notion of effective dimension was introduced [14] Recently, LUTZ [13] has extended the theory of effective dimension to complexity theory by introducing resource bounded ....

P. Billingsley. Ergodic theory and information. Wiley, 1965.

.... metric and asymptotic results concerning digits, various analogies of this kind were previously established, especially by Jager and de Vroedt [5] and Sal at [8] for real L uroth series, and by Ruban [17] for p adic continued fractions, in comparison with classical theorems of Khintchine (see e.g. [2], 6] for real continued fractions. In these developments, Haar measure for p adic numbers replaces Lebesgue measure for real numbers. The main aim of the present paper is to state or derive some similar metric and asymptotic results for the p adic L uroth type expansions referred to above. ....

Billingsley P, Ergodic Theory and Information. J. Wiley, 1965.

.... ) s: Besicovitch [1] proved that dimH(FREQ ) H( for all 2 [0; where FREQ = freq(S[0: n 1] Good [13] conjectured that the limit superior could be replaced by a limit here, thus obtaining dimH(FREQ ) H( for all 2 [0; 1] Eggleston [9] see also [2, 11]) proved Good s conjecture. The following corollary is a constructive version of Eggleston s theorem. Corollary 7.4. For all 2 [0; 1] cdim(FREQ ) H( Proof. This follows immediately from Lemma 7.3, Eggleston s above mentioned result, and Observation 3.7(2) Our second lemma gives an ....

P. Billingsley. Ergodic Theory and Information. John Wiley & Sons, 1965.

....proof. The details, for both the consistency theorems, are available through the technical report [26] The proof is based upon two lemmas. The first is a kind of uniform law of large numbers for the probabilities m , m n , reminiscent of the Shannon McMillan Breiman Theorem (cf. Billingsley [9]) Lemma 3.1.1 m#MNn log m (z 0 , z 1 , z n ) d o (z) 0 a.s. o ) su#ciently slowly. The second lemma insures that there is some sequence mN approaches o : Lemma 3.1.2 There exists a sequence of matrices mN N## , d o (z) ....

P. Billingsley. Ergodic Theory and Information. John Wiley & Sons, New York, 1964.

....is essential for systems with many degrees of freedom. The choice of the coarse grained variables is dictated by the nature of the system and by the questions of interest. One is then interesting in the amount of information lost in the coarse graining, at least in some statistical way [2]. In this paper we shall study this question for simple models of polymers, large molecules consisting of a linear sequence of N monomer units. A reduced description of this system can be based [7, 10, 13, 14] on associating to each polymer con guration a connectivity or contact matrix C, such ....

....(1.6) and P( jC) P( P(C)j C=C( 1.7) is the conditional probability of given that it is in C . We can thus think of SC as the average conditional entropy relative to C. Since SC 0 we clearly have SC S, and S SC is then a measure of information lost in the coarse graining [2]. The question is how much. In particular we may ask how does N SC =S (1.8) behave as N 1. Before answering this question we note that S C S C = ln jCj; 1.9) where SC is the entropy of the distribution SC which assigns equal weight to all C C , i.e. P(C) jCj , with jCj = ....

P. Billingsley, Ergodic Theory and Information, New York, Wiley ed. (1965).

....conclusions this paper. Finally, the proof of the algorithm convergence is presented in the Appendix. 2 Preliminaries In this section we present some definitions and notation that relate to sta tionary stochastic processes and Hidden Markov Models. The material is standard (see e.g. Billingsicy [7, 8]) We study discrete valued, stationary stochastic processes, e.g. t t 1, taking values in y = 1, 2, L . There is no loss of generality in assuming Yt to be integer valued. We define = Vl. VN, V, Y 1 n iV iV = l, 2, 4 has probability function p defined for all N O, ....

....offense: sometimes we write vo v ) s) p(vo Iv . v ) rrob(Vo = vo I yo = v , Now, strictly speaking, our processes are not defined for t O. However it is a simple matter to extend a one sided ( 0 t oo) stationary stochastic process to a two sided one ( c o t c o) see [7]) so the expression in eq. 8) is meaningful. Finally we define the entropy H(p) of a process p and the cross entropy H(q; p) of a process q with respect to a process p. H(p) f logp(Vo IV V 2. dw(voV . 9) H(q;p) f logq(vo l v v 2. dw(VoV . H(p) 10) These are formally ....

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P. Billingsley, Ergodic Theory and Information, Wiley, New York, 1965.

....is (E) 0 ) E) 1.16) In particular we have (A n ) l n (A l ) and (A n ) F 1 (B n ) B n ) where B n is as above. We then have (E) 1 ) E) 0 1 ) E) G E) 1.17) which says that is G invariant. Moreover is ergodic with respect to G (see e.g. [Bi]) Setting h(x) dx) dx we get h = j 1 j e 1 ; e = 0 ) h 0 ; 1.18) which gives the well known result h(x) 1 (1 x) 1.19) The primitive H(x) of h(x) with H(0) 0, is H(x) log(1 x) log 2. Setting q n : H( log 2) we have (A n ) ....

....K is de ned by K : 1.23) Proof. We have (log ) A k ) log k = q k 1 q k ) log k k 1 1 = K 1: This computation shows both that log 2 L 1 ( and the last equality in (1.22) The rst equality in (1. 22) now follows from the ergodic theorem [Bi]. The constant K which appears above is known in number theory as Khinchin s constant. This is not a coincidence, as we now brie y explain. 1.3 Connection with number theory The Farey sum over two rationals 0 is the mediant operation given by [HR] a a b b 0 (1.24) It ....

P Billingsley, Ergodic Theory and Information, John Wiley, 1964.

....stability property that dimFS (X [ Y ) max fdimFS (X) dimFS (Y )g for all X;Y C. We show that nite state dimension endows Q, the set of all binary expansions of rational numbers in [0; 1] with internal dimension structure. We show that the above mentioned theorem of Eggleston [5] see also [1]) holds for nite state dimension in both Q and C. In particular, Q itself has nite state dimension 1. For an individual sequence S 2 C, we de ne the nite state dimension of S to be dim FS (S) dimFS (fSg) Each element of Q has nite state dimension 0, while every sequence that is normal in ....

P. Billingsley. Ergodic Theory and Information. John Wiley & Sons, 1965.

....by Rieger [30] The related distribution functions are F S (x) 1 log 2 log 1 1 x ; FC (x) 1 log OE log OE(OE x) OE 2 Gamma x : We consider now some asymptotic mean values that play a central role in the sequel. The following results are classical and can be found for instance in [5]. Digit costs. Consider some particular digit cost c : M R such that c ffi M is in L 1 . If c(m) equals for instance the binary length of digit a, i.e. the number of binary digits in the binary expansion of digit a, one has, with general formula (20) A(S) E [S] 1 [ a) 1 ....

P. Billingsley, Ergodic Theory and Information, John Wiley & Sons (1965).

....dimension plays in fractal geometry, where it turned out to be a suitable concept for distinguishing fractal sets from sets of a rather smooth geometrical nature. In the context of computability, originating from applications of Hausdorff dimension in information theory (see for example [4]) a close connection between Hausdorff dimension and Kolmogorov complexity was established [6, 16, 17, 18, 20, 21] and the notion of effective dimension was introduced [14] Recently, LUTZ [13] has extended the theory of effective dimension to complexity theory by introducing resource bounded ....

P. Billingsley. Ergodic theory and information. Wiley, 1965.

.... R( H( 0 ) Since this holds for all rational 0 and H is continuous, it follows that dim(FREQ( j R( H( The case = all of Theorem 5. 3 says simply that the classical Hausdor dimensions of FREQ( and FREQ( are both H( This was proven in 1949 by Eggleston[5, 2]. The proof here is a new proof, using gales, of this classical result. However, it is complexity theoretic results of the following kind that are of interest in this paper. Corollary 5.4. 1. For all p computable reals 2 [0; 1] dim(FREQ( j E) dim(FREQ( j E 2 ) H( 2. For all 2 ....

P. Billingsley. Ergodic Theory and Information. John Wiley & Sons, 1965.

....binary) sequence S a dimension, which is a real number dim(S) in the interval [0; 1] Sequences that are random (in the sense of Martin L of) have dimension 1, while sequences that are decidable, 0 1 , or 0 1 have dimension 0. It is shown that for every 0 2 computable real number in [0,1] there is a 0 2 sequence S such that dim(S) A discrete version of constructive dimension is also developed using termgales, which are gale like functions that bet on the terminations of ( nite, binary) strings as well as on their successive bits. This discrete dimension is used to assign ....

.... c: 6 Given a set A f0; 1g and n 2 N, we use the abbreviations A=n = A f0; 1g n and A n = A f0; 1g n . A pre x set is a set A f0; 1g such that no element of A is a pre x of another element of A. Let X be a k fold product of intervals, each of which is (0,1) or [0,1]. If g : X R and = 1 ; k ) 2 X, then we sometimes use g( as an abbreviation for the random variable : f0; 1g R de ned by (1) g( 1 ; k ) and (0) g(1 1 ; 1 k ) If 2 [0; 1] then we also use as an abbreviation for the probability measure ....

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P. Billingsley. Ergodic Theory and Information. John Wiley & Sons, 1965.

....fx = x 1 : x n ; any n 1; x 1 ; x n 2 g (1.2) Deflnition 3 Given an alphabet , we deflne the set 1 as follows: 1 = fx = x 1 x 2 : any x 1 ; x 2 ; 2 g: 1.3) We assume the deflnitions of a stationary and ergodic stochastic process to be known. See, for example, Billingsley [Bil65] We also assume known the concept of a probability measure , or simply a probability [Bil76] This is to be distinguished from a probability function, which will be deflned presently. For a clariflcation of the difierence between a probability and a probability function, see the remarks below. A ....

....lim n 1 1 n X x2 n p(x) log p(x) H(p) 1.10) Proof: In Ash [Ash65] Theorem 3 (Shannon Breiman Macmillan) Given a stationary ergodic process X, with flnite alphabet and probability function p, we have lim n 1 1 n log p(X 1 : X n ) a:s = H(p) 1. 11) Proof: In Billingsley [Bil65] Similarly to entropy of a process we deflne the cross entropy of two processes: 21 Deflnition 11 Given two stationary stochastic processes X, Y with probability functions p, q respectively, and flnite alphabet , we deflne the n th order cross entropy of Y with respect X, denoted by Hn (Y ....

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P. Billingsley. Ergodic Theory and Information. Wiley, New York, 1965.

....dimension of set M with respect to measure is de ned as dim (M) inff : lim 0 (M; 0g = supf : lim 0 (M; 1g: Note that 0 dim (M) 1 for any and M . Note also that for the Lebesgue measure L, dimL (M) coincides with the Hausdor dimension in the usual sense [1], which is denoted by HD(M ) We de ne the mapping : B 1 [0; 1] associating with each x 1 1 2 B 1 the real number (x 1 1 ) the binary expansion 0:x 1 x 2 . For each I B 1 , we denote as HD(I) the Hausdor dimension of set (I) 3] Then, the following facts are shown in ....

....paper, we prove the following theorem: Theorem 1: HD(K( nM( 2. Proof of Theorem First, we state two lemmas needed for the proof. Lemma 1: 6] 7] 1. A 1 A 2 ) dim (A 1 ) dim (A 2 ) 2. A) 0 ) dim (A) 1; and 3. dim ( n An ) sup n fdim (An )g. Lemma 2: [1]) Let and be probability measures. Then, M fx 1 1 j lim inf n 1 log (x n 1 ) log (x n 1 ) g implies dim (M) dim (M ) 2 IEICE TRANS. FUNDAMENTALS, VOL.E82 , NO.1 JANUARY 1999 (Proof of theorem 1 ) HD(K( nM( is clear because HD(K( and K( nM( ....

P. Billingsley, \Ergodic Theory and Information," John Wiley & Sons, 1965.

....lim n 1 (T n A) 1 for any A 2 B(X) with (A) 0: All these basic notions arise in ergodic theory. It can be shown that exact ) mixing ) weakly mixing ) ergodic; none of these implications being an equivalence. For proofs, examples and counterexamples, we refer the reader to Billingsley [3], Lasota and Mackey [8] Petersen [9] or Pollicott and Yuri [10] 2 Sensitive endomorphisms In this section, we consider a given endomorphism T on the metric probability space (X; d; B(X) Recall (see Pollicott and Yuri [10] page 101) that a set of integers N ae N is said to have positive ....

Billingsley, P. (1965) Ergodic Theory and Information, Wiley Series in Probability and Mathematical Statistics, New York.

....class of measures that we shall denote by D a; or, shortly, D a . This class does not depend on the Markov partition neither on the riemannian volume used in the construction. D a is a metric outer measure and it restricts to a regular measure on the oe algebra of Borel sets. See [8] [5]. Definition 5.2 The dynamical dimension of a Borel subset X is dimD (X) inffa 0 : D a; X) 0g Let us underline that these definitions are a particular case of a general construction due to P. Billingsley. cf. 4] and [5] The following outer measure is equivalent to D a; and will be ....

....to a regular measure on the oe algebra of Borel sets. See [8] 5] Definition 5.2 The dynamical dimension of a Borel subset X is dimD (X) inffa 0 : D a; X) 0g Let us underline that these definitions are a particular case of a general construction due to P. Billingsley. cf. 4] and [5]. The following outer measure is equivalent to D a; and will be used to describe some of their properties: D a; A) sup n 0 inf kn X P2(k) P A6= Vol (P ) a : In general D a D a since we are taking infimum over a smaller class of coverings. Nevertheless, it can be proved that ....

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Billingsley, P.: Ergodic Theory and Information. Wiley & Sons. 1965.

....t it lifts f, since ffi F = f ffi , where : Z Gamma X is the projection onto the first factor (z) x 0 . Indeed, the shift oe(z) n = z n 1 is the inverse map of F as it is easy to verify. There is a 1 to 1 and onto correspondence Gamma between the space of Borel probability measures in [0,1] M = M( 0; 1] and M = M(Z) such that f : 0; 1] B; becomes isomorphic to F : Z; B; cf. 26] We include in the Appendix for a proof of this a related facts for the completeness of the exposition. We shall use repeatly the well known features of the measurable decompositions of a ....

....n 0 and define the decomposition Xi = 1 n=0 F Gamman Sn : 10) Let S = W 1 n=0 F Gamman S. Then, by definition ( Xi(z) S ( z) Consequently, two pre orbits z and z 0 belong to the same Xi atom if and only if fxn gn0 and fx 0 n gn0 have the same S itinerary in [0,1]: S(xn ) S(x 0 n ) 8 n 0: Xi is a measurable decomposition. For every z 2 Z the projection ( Xi(z) 0; 1] is a closed interval or a point. If has positive Lyapunov exponent and satisfies some regularity condition there is a family of local unstable manifolds forming a measurable ....

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Billingsley, P.: Ergodic Theory and Information. Wiley & Sons. 1965.

....For any measure , let (27) dim def = inf dim H S SisaBorelsetwith(S) 1 . Define the lower dimension of by (28) dim def = inf dim H S SisaBorelsetwith(S) 0 . ENTROPY OF CONVOLUTIONS ON THE CIRCLE 897 To compute dimension of measures we use the following lemma from Billingsley [1]. Billingsley s Lemma. Let be a positive finite measure on T. Assume K # T is a Borel set satisfying (K) 0 and K # x # T : lim inf ##0 log [B # (x) log # # # . Then dimH K # #. If the lim inf is # a.e. then dimH K = #. Here B # (x) can be the interval of length # centered ....

P. Billingsley, Ergodic Theory and Information, John Wiley & Sons, Inc., New York, 1965.

.... of nice subsets of A 1 (considered as subsets of [0; 1] Or, if one considers arbitrary subsets of A 1 , the entropy H might be interpretated as a form of the box counting dimension, see e.g. Falconer, 1990] The connection between entropy and Hausdor dimension is described in [Billingsley, 1965] and interesting results in this direction can be found in [Ryabko, 1986] ....

Billingsley, P. Ergodic theory and Information. John Wiley & Sons, 1965.

....state 1 leads deterministically to state 2 and the transition probabilities from state 2 to state i 2 f1; 2g are proportional to C i ( given in (2. 3) This follows from the well known relation between Holder exponents and Hausdorff dimension and the Shannon McMillan Breiman Theorem; see, e.g. Billingsley (1965). Thus, we find that dim( T ) 1 p 1 2 p 1 log(1 p 1) Gamma p 1 (2 p 1) log p 1 : This is easily shown to be monotonic for 0 ( p 5 1) 2 and quite close to constant, going from 2(log 2) 3 0:46 up to log Gamma ( p 5 1) 2 Delta 0:48. Figure 5 ....

Billingsley, P. (1965). Ergodic Theory and Information. Wiley, New York.

....j 1 ; ffl j k j fx j : x j 1 = ffl j 1 ; x j k = ffl j k g, where ffl j 2 f0; 1g. Indicator functions for such cylinder sets (that is, functions which have value 1 on the set and value 0 off the set) are continuous. Now when X is used as the model for the history of flips of a fair coin [Bil], it is natural to associate with the cylinder set X ffl j 1 ; ffl j k an expected value (1=2) k , which would be the value of the integral I( X ffl ) of the indicator function X ffl . In general, an expected value or (probability) integral I would have the properties: ffi I(f) 0 for ....

P. Billingsley, Ergodic Theory and Information, John Wiley, New York, 1965.

....properties which are global in the sense of pertaining to a range of winding numbers. We first briefly review some of the attempts to derive such predictions using ideas from the ergodic number theory, and then turn to the predictions based on the thermodynamic foramlism. The ergodic number theory[25, 42] is rich in (so far unfulfilled) promise for the mode locking problem. For example, while the Gauss shift (6) invariant measure (x) 1 ln 2 1 1 x (19) was known already to Gauss, the corresponding invariant measure for the critical circle maps renormalization operator R has so far eluded ....

P. Billingsley, Ergodic Theory and Information (Willey, New York 1965)

....of I ae C 1 for ae 0 if 1. 8x 2 I has 9oe 2 S as its unique prefix in S; and 2. j(oe) ae for 8oe 2 S. Let Gamma(I ; ae) denote the set of such S s, and l (ff) j (I ; ae) inf S2 Gamma(I;ae) X oe2S j(oe) ff : Then, the Hausdorff dimension of I with respect to j is defined by [1] dim j (I) inffff : lim ae 0 l (ff) j (I ; ae) 0g = supfff : lim ae 0 l (ff) j (I ; ae) 1g : 3. Main Result Let A and B be finite sets, and I ae A 1 . Let and be probability measures over A and B , respectively. We define Gamma1 (y) fx 2 I : y OE (x)g for y 2 ....

P. Billingsley, Ergodic theory and information, John Wiley & Sons (1965).

....shift. Finally, T is ergodic under Lebesgue measure (and in fact is mixing) Proof: It is easily checked that T corresponds to the shift map of S, i.e. if x = 1 a1 1 a1 a2 ; then Tx = a 1 x 1. The proof that T is ergodic is analogous to the one for continued fractions, e.g. see [3]. 2 Corollary 2. The ergodic theorem implies that, with probability one, lim n 1 n p a 1 a 2 an = 2 1 k=1 k=2 k = 4 ; the analogue of Khinchin s constant [14] 3) lim n 1 a t 1 a t 2 a t n n 1=t = 1 X k=1 2 kt 2 k 1=t = 2 1 t 1) 1=t ....

P. Billingsley, Ergodic Theory and Information, Wiley 1965.

....one learns about x. Thus, if one knows n base 10 digits of x taken in (0; 1) one has reduced x to an interval of size 10 n . Knowledge of one more digit 12 further reduces the interval to size 10 n 1 , therefore, one has narrowed x by a factor of 10. The Shannon McMillan Breiman theorem [5] then says that the entropy of the shift map T 10 ( a 1 a 2 : a 2 a 3 : is log 10. In the case of continued fractions, one looks at an x 2 (0; 1) and writes x = 0; a 1 (x) a 2 (x) The rst n continued fraction digits give an approximation pn q n = 0; a 1 (x) an ....

.... log 2 N one might expect that SN N log 2 N a.e. but this is false in general. In fact, a result of Borel and Bernstein states that if (1) 2) is a sequence of positive integers then for almost all x 2 (0; 1) an (x) n) in nitely often if and only if P 1= n) diverges (see [5] for a proof) In particular, one has an (x) n log n log log n in nitely often, for almost all x. 17 Another diculty is that the an (x) are not independent, and in fact this theory is one of the fundamental examples of sums of identically distributed non independent random variables. In spite ....

P. Billingsley, Ergodic Theory and Information, Wiley, New York 1965.

....g 1 k=1 is a stationary ergodic sequence of symbols generated from a finite alphabet Sigma, but this is too strong for our purpose. Therefore, we adopt the following two weaker probabilistic models. A1) Mixing Model The sequence fX k g 1 k= Gamma1 satisfies the so called mixing condition [5], that is, there exist two constants 0 c 1 c 2 and an integer d such that for all Gamma1 m m d n the following holds c 1 PrfAgPrfBg PrfABg c 2 PrfAgPrfBg (2:4a) with A 2 F m Gamma1 and B 2 F 1 m d where F n m is a oe field generated by fX k g n k=m for m n. In some statements ....

....needed for the formulation of our results. First of all, let X n m = Xm ; X n ) for m n, and let for every n 1 the nth order probability distribution for fX k g be P (X n 1 ) PrfX k = x k ; 1 k n; x k 2 Ag : Then, the entropy of fX k g is defined in a standard manner as (cf. [5]) h = lim n 1 E log P Gamma1 (X n 1 ) n : The next three parameters are well defined under our assumption (A1) cf. 10] 30] Definition 3. Renyi s order entropy For Gamma1 b 1, define the bth order R enyi entropy as h (b) 2 = lim n 1 log(EfP b (X n 1 )g) Gamma1 (b ....

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P. Billingsley, Ergodic Theory and Information, John Wiley & Sons, New York (1965).

.... Gamma 1 N log( I N oe(E) 1 N P N Gamma1 i=0 log(jS 0 (S ffii (E) j) d (E) lim sup N 1 Gamma 1 N log( I N oe(E) 1 N P N Gamma1 i=0 log(jS 0 (S ffii (E) j) It immediately follows from the Breiman Shannon McMillan theorem and Birkhoff s ergodic theorem [5], that the upper and lower pointwise dimension coincide, and are constant almost surely whenever the measure is ergodic. This gives formula (2) 15 For the proof of Theorem 3, we shall need several reformulations of the basic contraction properties of S stating that there exist finite ....

P. Billingsley, Ergodic Theory and Information, (Wiley, New York, 1965).

....: lim r#0 log 1 (B r (x) OE log 1 r (4:1) when the limit of the above quotient exists. Example: For a Borel probability measure on T , we have Ho( lim n 1 1 n log 1 ( n ) The relationship of Holder exponent to Hausdorff dimension is given in the following result of Billingsley (1965), x14; see also Young (1982) Billingsley proved a more general result for euclidean space, but the same proof works even more easily on the boundaries of trees. Lemma 4.1. For any Borel probability measure on the boundary of a tree, if the Holder exponent of exists a.e. and is constant, ....

Billingsley, P. (1965). Ergodic Theory and Information. Wiley, New York.

....Holder exponent of at a point x as lim r#0 log 1= B r (x) log 1=r ; where B r (x) denotes the ball of radius r centered at x if this limit exists. For example, if is Lebesgue measure on n dimensional euclidean space, then it is clear that its Holder exponent is n at every point. Billingsley (1965) showed that if the Holder exponent exists and is constant a.s. then it equals dim . He showed this for euclidean space, but the same proof works even more easily on the boundary of trees. 11 Figure 12. The difference between limit uniform and harmonic measures. If 2 T is the ray h 0 ; 1 ....

Billingsley, P. (1965). Ergodic Theory and Information. Wiley, New York.

....state 1 leads deterministically to state 2 and the transition probabilities from state 2 to state i 2 f1; 2g are proportional to C i ( given in (2. 3) This follows from the well known relation between Holder exponents and Hausdorff dimension and the Shannon McMillanBreiman Theorem; see, e.g. Billingsley (1965). Thus, we find that dim( T ) 1 p 1 2 p 1 log(1 p 1) Gamma p 1 (2 p 1) log p 1 : This is easily shown to be monotonic for 0 ( p 5 1) 2 and quite close to constant, going from 2(log 2) 3 0:46 up to log Gamma ( p 5 1) 2 Delta 0:48. ....

Billingsley, P. (1965). Ergodic Theory and Information. Wiley, New York.

....of order j containing x. Lochs [L] proved the following theorem: Theorem 1. Let denote Lebesgue measure on [0; 1) Then for a.e. x 2 [0; 1) lim n 1 m(n; x) n = 6 log 2 log 10 2 : Lochs proof was based on the intricate arithmetic properties of the RCF map, and on a result by L evy [B] which states that lim n 1 Qn n = 2 12 log 2 a.e. where Pn Qn = 0; a 1 ; a 2 ; Delta Delta Delta ; an ] In 1999, Bosma Dajani and Kraaikamp [BDK] generalized theorem 1 to a wider class of transformations on [0; 1) by noticing that Lochs s theorem is concerned with the way the ....

....a wider class of transformations on [0; 1) by noticing that Lochs s theorem is concerned with the way the decimal cylinders fit in the CF cylinders, and that the limit is in fact the ratio of the entropies of the maps under consideration. Their proof was based on Shannon McMillan Breiman Theorem [B], and the dynamics of the underlying transformations as reflected in the way the partitions are refined under iterations of the corresponding maps. A surjective map T : 0; 1) 0; 1) is called a number theoretic fibered map (NTFM) if it satisfies the following conditions: a) there exists a ....

Billingsley, P. -- Ergodic Theory and Information, John Wiley and Sons, Inc.; New YorkLondon -Sydney (1965).

....distribution q. We denote by q the distribution of X itself and by S the shift operator on bi infinite sequences, i.e. for x = x n ) n2Z we have Sx : x n 1 ) n2Z . Then i ]0; 1[ Z ; B; q; S j , where B is the completion of the Borel field, is a measure preserving system (see [3]) which we assume to be ergodic. Note that this notion of ergodicity differs from the one used for Markov chains. So, for example, we do not rule out periodic chains. For a Markov chain with finite state space, we only need the transition matrix to be irreducible. We interpret X as a random ....

....and x EX 0 . In the following we introduce a counter example to the statement that a random walk in a more random environment escapes more slowly than a random walk in a less random environment. We are going to measure randomness by entropy, and we will consider Markov chain environments. By [3], the entropy, i.e. the amount of randomness, of a Markov chain with finitely many states, stationary distribution q and transition matrix (q ij ) is given by E = Gamma X i;j q i q ij log q ij : Let the random environment (Xn ) n2Z be a Markov chain with state space fp 1 ; p 2 g Z = ae ....

Billingsley, P. (1965). Ergodic Theory and Information , Wiley, New York

....OE t 1 OE 2 Gamma t ] The related distribution functions are FS (x) 1 log 2 log 1 1 x ; FC (x) 1 log OE log OE(OE x) OE 2 Gamma x : We consider now some asymptotic mean values that play a central role in the sequel. The following results are classical and can be found in [5]. Digit costs. Consider some particular digit cost c : M R such that c ffi M is in L 1 . If c(m) equals for instance the binary length of digit a, i.e. the number of binary digits in the binary expansion of digit a, one has, with general formula (20) A(S) E [S] 1 [ a) 1 log 2 ....

Billingsley, P. Ergodic Theory and Information, John Wiley & Sons, 1965.

....cases. A mathematical theory for the information content of a source of messages, or a transmission channel was formulated around the middle of the 20 th century by Shannon, Kolmogorov, and a number of other main figures. Original material can be found in Refs. 7, 8] a good text book is Ref. [9]. Shannon s classical text is surprisingly readable and a good paperback edition exists. Let us briefly recall the most basic concepts. Suppose a discrete variable I is drawn according to a probability distribution p(i) The average number of bits needed to optimally encode each outcome is given ....

P. Billingsley. Ergodic Theory and Information. Wiley, New York, 1965.

....cases. A mathematical theory for the information content of a source of messages, or a transmission channel was formulated around the middle of the 20 th century by Shannon, Kolmogorov, and a number of other main figures. Original material can be found in Refs. 16, 17] a good text book is Ref. [18]. Shannon s classical text is surprisingly readable and a good paperback edition exists. Let us briefly recall the most basic concepts. Suppose a discrete variable I is drawn according to a probability distribution p(i) The average number of bits needed to optimally encode each outcome is given ....

P. Billingsley. Ergodic Theory and Information. Wiley, New York, 1965.

....1 K # ( m 0 m d 1 ) 1 d d . More generally, if limiting frequencies f 0 , f d 1 exist (mod d) for the trajectory T K (x) then T K (x) 1 K # m 0 f 0 m d 1 f d 1 d . 7 5 Ergodic theory One innovation in [14] was the introduction of ergodic theory (see [1]) The mapping T extends uniquely to a continuous mapping of the d adic integers Z d into itself. Z d can be regarded as a completion, consisting of formal sums # # i=0 a i d i , a i # 0, 1, d 1 , with addition and multiplication done as with ordinary positive integers, ....

P. Billingsley, Ergodic theory and information, John Wiley, New York 1965.

....If Q 2 Gamma Q 1 is integrable, then E(Q n 1 Gamma Q n ) 0, even when the Q n s are not integrable. Proof: Because (Q n ) n is stationary, there exists a shift on the canonical probability space of the sample paths of the process (Q n ) n which preserves the probability measure (see Billingsley [1965], p.19) To avoid additional notations, we suppose that our variables are defined on this canonical probability space. By the stationarity, it suffices to prove that E(Q 2 Gamma Q 1 ) 0. Let I be the oe field of the invariant events. Put Q = Q 1 , and define for any constant ffi the integrable ....

Billingsley, P. [1965],Ergodic Theory and Information, John Wiley & Sons.

....initial distribution F , while the main terms are independent of the initial distribution and depend only on the mechanism S of the source. This result is a strong form of the Almost Equipartition Property, known as the Shannon MacMillan Breiman Theorem that can be described as follows: see [3], 32] 41] for more details) Theorem. Shannon MacMillan Breiman Theorem] Let S a stationary ergodic source with entropy h(S) and alphabet M. Then, for any 0, there exists a positive integer K 0 ( such that, if k K 0 ( the set M k of prefixes of length k decomposes into two ....

Billingsley, P. Ergodic Theory and Information John Wiley & Sons, 1965.

....property in information theory is called the chain rule. Thus, H n ) n2N is the discrete derivative of (V n ) n2N . Note that (V n ) n2N is a non decreasing sequence, since H n 0 for all n. It can be shown that (H n ) n2N is a monotonic non increasing sequence of n (see, for instance [16]) The intuitive meaning of this is that the uncertainty about the choice of the next symbol decreases when the number of known preceding symbols increases. From the non increasing behaviour of the positive sequence (H n ) n2N , we deduce the existence of the limit lim n 1 H n . We have: 8n, H ....

P. Billingsley, Ergodic Theory and Information, John Wiley and Sons, New York (1965).

.... of M ae [0; 1] Then, the Hausdorff dimension of set M with respect to measure is defined as dim (M ) inffff : lim ae 0 ff (M; ae) 0g = supfff : lim ae 0 ff (M; ae) 1g : Note that for Lebesgue measure L, dimL (M ) coincides with the Hausdorff dimension in the usual sense [1], which is denoted by DH(M ) Concerning Hausdorff dimension, the following properties are known [1] 1. M 1 ae M 2 ) dim (M 1 ) dim (M 2 ) 2. M ) 0 ) dim (M ) 1; and 3. dim ( n Mn ) sup n dim Mn . Definition 4 We define the mapping : B 1 [0; 1] associating with each x ....

.... as dim (M ) inffff : lim ae 0 ff (M; ae) 0g = supfff : lim ae 0 ff (M; ae) 1g : Note that for Lebesgue measure L, dimL (M ) coincides with the Hausdorff dimension in the usual sense [1] which is denoted by DH(M ) Concerning Hausdorff dimension, the following properties are known [1]; 1. M 1 ae M 2 ) dim (M 1 ) dim (M 2 ) 2. M ) 0 ) dim (M ) 1; and 3. dim ( n Mn ) sup n dim Mn . Definition 4 We define the mapping : B 1 [0; 1] associating with each x 1 1 2 B 1 the real number (x 1 1 ) the binary expansion 0:x 1 x 2 : For each I ae B ....

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P. Billingsley, Ergodic Theory and Information, Wiley, 1965

....function f on X; a Borel measure on X is called Gibbs with respect to the potential f , if there exists a constant C 1 such that, for all x = x 1 ; x 2 ; 2 X and n 1 , C Gamma1 Delta exp( GammaS n f(x) x 1 ; x n ] C Delta exp( GammaS n f(x) Definition 2. cf. [1]) For a finite measure on X; F ae X; s 0; ae 0 , let M s (F ) lim ae#0 M s ;ae (F ) where M s ;ae (F ) inf n P z2W (z) s : W ae Z s.t. F ae S z2W z; z) ae 8z 2 W o : Then dim (F ) the Hausdorff dimension of F with respect to , is defined by dim (F ) inf ....

....it is always true that dim (F ) 1: Furthermore, if (F ) 0; then 0 M 1 (F ) 1 and thus dim (F ) 1: We now turn to the proof of Theorem 2, where we refer to the introduction for the actual statement of this theorem. Proof of Theorem 2. Recall the following result of Billingsley ([1]) which states that if m 1 ; m 2 are finite measures on X; and if # 0 and F ae X are such that lim n 1 log m 1 [ x 1 ; x n ] log m 2 [ x 1 ; x n ] # for all x = x 1 ; x 2 ; 2 F ; then dimm 2 (F ) # Delta dimm 1 (F ) Now let D ae X , and fix a oe invariant ergodic ....

[Article contains additional citation context not shown here]

P. Billingsley, Ergodic Theory and Information, Wiley & Sons, New York, 1965.

....Note that for z 1 2 this is an empty set. Finally in this section we consider the inequality fi fi fi fi x Gamma pn q n fi fi fi fi 1 2q n q n Gamma1 : n = 1; 2; This is sharper than (1.1) whenever c n = 1. That is for almost all x with frequency 2 Gamma log 3 log 2 . See [B] for details. This motivates the following theorem. Metric Diophantine Approximation 123 Theorem 4.8. For each irrational number x in (0; 1) define the function Dn (x) for each natural number n by the identity fi fi fi fi x Gamma pn q n fi fi fi fi = Dn (x) q n q n Gamma1 : n = 1; 2; ....

P. Billingsley, Ergodic Theory and Information, John Wiley and Sons, New York, 1965.

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P. Billingsley. Ergodic Theory and Information. John Wiley & Sons, 1965.

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P. Billingsley, Ergodic theory and information, John Wiley & Sons (1965).

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P. Billingsley. Ergodic Theory and Information. John Wiley & Sons, New York, London, Sydney, Wiley Series in Probability and Mathematics edition, 1965.

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P. Billingsley, Ergodic Theory and Information, John Wiley & Sons, Inc., New York, 1965.

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P. Billingsley, Ergodic Theory and Information, John Wiley, New York, 1965.

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P. Billingsley (1965), Ergodic Theory and Information, Wiley, New York.

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