On the central limit theorem for Markov chains

В. A. Lifšic

Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 2, pp. 295-312

Résumé

Let, for each $n=1,2,\dots$, random variables $X_{ns}$, $1\le s\le n$, form a (non-homogeneous) Markov chain, $\mathbf EX_{ns}=0$. Let $\mathscr B_{ns}$ be the $\sigma$-algebra generated by $X_{ns}$ and $\beta_{nt}$ be the maximal correlation coefficient between $\mathscr B_{nt}$ and $\mathscr B_{n,t+1}$. Denote \begin{gather*} S_n=\sum_sX_{ns},\quad F_n(x)=\mathbf P\{S_n<x\sqrt{\mathbf DS_n}\},\\ F_{ns}(x)=\mathbf\{X_{ns}<x\},\quad\beta_n=\max_t\beta_{nt}. \end{gather*} Theorem 3. {\it If $0<c<\mathbf DX_{ns}<C<\infty$ and, for each $r>0$, $$ \frac{1}{n(1-\beta_n)^2}\sum_s\int_{|y|>y\sqrt n(1-\beta_n)^{3/2}}y^2F_{ns}(dy)\to 0,\ n\to\infty, $$ then $F_n(x)$ converges to the standard normal distribution function.} We also consider (in Theorem 9) the case of a stationary Markov chain under condition $\beta_n=1$ ($n=1,2,\dots$).