Chung's law for additive functionals of positive recurrent Markov chains (Q1976507)

Consider a positive recurrent Markov chain \(\{ X_n\}\) with stationary measure \(\pi\) and transition probabilities \(P\). Further, take \(f\) to be a real-valued function with zero mean and finite variance (integrated against \(\pi\)) and for which \(\sum_{1}^{\infty}f(\cdot)P^k f(\cdot)\) converges in \(L_1(\pi)\). The paper establishes Chung's lim inf LIL for \(\{ f(X_i) \}\), i.e. with probability one \[ \liminf_{n\rightarrow\infty} \left\{((\log\log n)/n)^{1/2}\max_{k\leq n} \left|\sum_{j=1}^{k}f(X_j)\right|\right\} =\pi\sigma_f/8^{1/2}, \] where \[ \sigma_f^2=\int f^2(x)\pi(dx) + 2\sum_{k=1}^{\infty}\int f(x)P^kf(x)\pi(dx). \] As a straightforward regeneration argument is not sufficient, the author employs a minorization based on Theorem 16.02 of \textit{S. P. Meyn} and \textit{R. L. Tweedie} [``Markov chains and stochastic stability'' (Springer, 1993)] to complete the proof.