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## Colloquium Mathematicum

2000 | 84/85 | 1 | 159-171
Tytuł artykułu

### Average convergence rate of the first return time

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The convergence rate of the expectation of the logarithm of the first return time $R_{n}$, after being properly normalized, is investigated for ergodic Markov chains. I. Kontoyiannis showed that for any β > 0 we have $log[R_{n}(x)P_{n}(x)] =o(n^{β})$ a.s. for aperiodic cases and A. J. Wyner proved that for any ε >0 we have $-(1 + ε)log n ≤ log[R_{n}(x)P_{n}(x)] ≤ loglog n$ eventually, a.s., where $P_{n}(x)$ is the probability of the initial n-block in x. In this paper we prove that $E[log R_{(L,S)} - (L-1)h]$ converges to a constant depending only on the process where $R_{(L,S)}$ is the modified first return time with block length L and gap size S. In the last section a formula is proposed for measuring entropy sharply; it may detect periodicity of the process.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
159-171
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-06-11
poprawiono
2000-01-18
Twórcy
autor
• Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon, South Korea
autor
• Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon, South Korea
Bibliografia
• [1] A. Dembo and I. Kontoyiannis, The asymptotics of waiting times between stationary processes, allowing distortion, Ann. Appl. Probab. 9 (1999), 413-429.
• [2] M. Kac, On the notion of recurrence in discrete stochastic processes, Bull. Amer. Math. Soc. 53 (1947), 1002-1010.
• [3] J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer, New York, 1976.
• [4] I. Kontoyiannis, Asymptotic recurrence and waiting times for stationary processes, J. Theoret. Probab. 11 (1998), 795-811.
• [5] I. Kontoyiannis, P. H. Algoet, Yu. M. Suhov and A. J. Wyner, Nonparametric entropy estimation for stationary processes and random fields, with applications to English text, IEEE Trans. Inform. Theory 44 (1998), 1319-1327.
• [6] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, 1995.
• [7] G. Louchard and W. Szpankowski, On the average redundancy rate of the Lempel-Ziv code, IEEE Trans. Inform. Theory 43 (1997), 2-8.
• [8] U. Maurer, A universal statistical test for random bit generators, J. Cryptology 5 (1992), 89-105.
• [9] D. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory 39 (1993), 78-83.
• [10] C. Shannon, The mathematical theory of communication, Bell Sys. Tech. J. 27 (1948), 379-423 and 623-656.
• [11] P. C. Shields, The Ergodic Theory of Discrete Sample Paths, Grad. Stud. Math. 13 Amer. Math. Soc., 1996.
• [12] W. Szpankowski, Asymptotic properties of data compression and suffix trees, IEEE Trans. Inform. Theory 39 (1993), 1647-1659.
• [13] F. M. J. Willems, Universal data compression and repetition times, ibid. 35 (1989), 54-58.
• [14] A. J. Wyner, Strong matching theorems and applications to data compression and statistics, Ph.D. thesis, Stanford Univ., 1993.
• [15] A. J. Wyner, More on recurrence and waiting times, Ann. Appl. Probab., to appear.
• [16] A. J. Wyner and J. Ziv, Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression, IEEE Trans. Inform. Theory 35 (1989), 1250-1258.
• [17] J. Ziv and A. Lempel, A universal algorithm for sequential data compression, ibid. 23 (1977), 337-343.
• [18] J. Ziv and A. Lempel, Compression of individual sequences via variable rate coding, ibid. 24 (1978), 530-536.
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Bibliografia
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