Convergence of moments in the central limit theorem for nonstationary Markov chains

В. A. Lifšic

Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 4, pp. 755-771

Résumé

Let, for every $n=1,2,\dots,$ random variables $X_{ns}$, $1\le s\le n$, form a Markov chain with transition functions $Q_{nt}$, $1\le t\le n-1$. We denote $$ S_n=\sum_sX_{ns},\quad F_n(x)=\mathbf P(S_n<x\sqrt{\mathbf DS_n}),\quad\alpha_n=\min_t\alpha(Q_{nt}), $$ where $\alpha(Q_{nt})$ is the ergodicity coefficient of $Q_{nt}$. Theorem. {\em If $$ |X_{ns}|\le C,\quad\mathbf EX_{ns}=0,\quad\mathbf DX_{ns}\ge c,\quad\alpha_nn^{1/3}/\ln n\to\infty, $$ then, for every $p\ge0$, $$ \int_{-\infty}^\infty|x|^pF_n(dx) $$ converges to the pth absolute moment of} $N(0,1)$.