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 $log\left[R_n\left(x\right)P_n\left(x\right)\right]=o\left(n^{\beta }\right)$ a.s. for aperiodic cases and A. J. Wyner proved that for any ε >0 we have $-(1+\epsilon )logn\le log\left[R_n\left(x\right)P_n\left(x\right)\right]\le loglogn$ eventually, a.s., where $P_n\left(x\right)$ 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...