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Average convergence rate of the first return time

100%

Choe G. H., Kim D. H.

Colloquium Mathematicum

2000

tom 84/85

nr 1

159-171

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Choe G. H.

1 Kim D. H.

1 2000

Wyniki wyszukiwania

Average convergence rate of the first return time