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Average convergence rate of the first return time

Geon Choe, Dong Kim (2000)

Colloquium Mathematicae

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 l o g [ 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 ) l o g n l o g [ R n ( x ) P n ( x ) ] l o g l o g 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 [ l o g 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...

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