# Average convergence rate of the first return time

Colloquium Mathematicae (2000)

- Volume: 84/85, Issue: 1, page 159-171
- ISSN: 0010-1354

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topChoe, Geon, and Kim, Dong. "Average convergence rate of the first return time." Colloquium Mathematicae 84/85.1 (2000): 159-171. <http://eudml.org/doc/210794>.

@article{Choe2000,

abstract = {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.},

author = {Choe, Geon, Kim, Dong},

journal = {Colloquium Mathematicae},

keywords = {entropy; the first return time; period of an irreducible matrix; Wyner-Ziv-Ornstein-Weiss theorem; data compression; Markov chain; ergodic Markov chains; modified first return time},

language = {eng},

number = {1},

pages = {159-171},

title = {Average convergence rate of the first return time},

url = {http://eudml.org/doc/210794},

volume = {84/85},

year = {2000},

}

TY - JOUR

AU - Choe, Geon

AU - Kim, Dong

TI - Average convergence rate of the first return time

JO - Colloquium Mathematicae

PY - 2000

VL - 84/85

IS - 1

SP - 159

EP - 171

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/210794

ER -

## References

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