Strong approximation for additive functionals of geometrically ergodic Markov chains

Abstract

Let $(\xi_i)_{i \in {\mathbb Z}}$ be a stationary Harris recurrent geometrically erodic Markov chain on a countably generated state space $(E, {\mathcal B})$. Let $f$ be a bounded and measurable function from $E$ into ${\mathbb R}$ satisfying the condition ${\mathbb E} ( f ( \xi_0))=0$. In this paper, we obtain the almost sure strong approximation of the partial sums $S_n(f) = \sum_{i=1}^n f ( \xi_i)$ by the partial sums of a sequence of independent and identically distributed Gaussian random variables with the optimal rate $O ( \log n)$.