Average convergence rate of the first return time

The convergence rate of the expectation of the logarithm of the first return time ${\displaystyle R_n}$, after being properly normalized, is investigated for ergodic Markov chains. I. Kontoyiannis showed that for any β > 0 we have ${\displaystyle \log \left[R_n\left(x\right)P_n\left(x\right)\right]=o\left(n^{\beta }\right)}$ a.s. for aperiodic cases and A. J. Wyner proved that for any ε >0 we have ${\displaystyle -(1+\epsilon )\log n\le \log \left[R_n\left(x\right)P_n\left(x\right)\right]\le \log \log n}$ eventually, a.s., where ${\displaystyle P_n\left(x\right)}$ is the probability of the initial n-block in x. In this paper we prove that ${\displaystyle E\left[{\log R}_{(L,S)}-(L-1)h\right]}$ converges to a constant depending only on the process where ${\displaystyle R_{(L,S)}}$ 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.