Average convergence rate of the first return time

• Autorzy: Geon Ho Choe , Dong Han Kim
• 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

entropy; the first return time; period of an irreducible matrix; Wyner-Ziv-Ornstein-Weiss theorem; data compression; Markov chain

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2000
• Tom: 84/85
• Numer: 1
• Strony: 159-171

Daty

wydano

2000

otrzymano

1999-06-11

poprawiono

2000-01-18

Twórcy

autor

Geon Ho Choe

• Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon, South Korea

autor

Dong Han Kim

• Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon, South Korea

Bibliografia

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