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

В. A. Lifšic

Leningrad

Abstract: 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} $N(0,1)$.

Received: 27.05.1974

 English version: Theory of Probability and its Applications, 1976, 20:4, 741–758

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