Some Limit Theorems for Stationary Markov Chains

Let X be a space of points, $F_X $ a $\sigma $-algebra of its subsets, and $p(\xi ,A)$, $\xi \in X$, $A \in F_X $, a stochastic transition function satisfying the following condition:an integer $k \geqq 1$ exists such that \[ {\text{(1)}}\qquad \mathop {\sup }\limits_{\eta ,\xi \in X,A \in F_X } \left| {p^{(k)} (\xi ,A) - p^{(k)} (\eta ,A)} \right| < 1. \] Let us define the sequence of random variables $x_1 ,x_2 , \cdots ,x_n , \cdots$ as follows: \[ \Pr \left( {x_1 \in A_1 ,x_2 \in A_2 , \cdots ,x_n \in A_n } \right) = \int\limits_{A_1 } {\pi (d\xi _1 )} \int\limits_{A_2 } {p(\xi _1 ,d\xi _2 ) \cdots } \int\limits_{A_n } {p(\xi _{n - 1} ,d\xi _n )} , \] where $\pi ( \cdot )$ is the initial distribution.Let $f(\xi )$ be a real function of $\xi \in X$ measurable with respect to $F_X $.In Chapter I the asymptotic behaviour of the characteristic function of $\sum _1^n f(x_i )$ is studied. Chapter II is devoted to limit theorems. The central limit theorem is proved under the assumption that \[ {\text{(2)}}\qq...