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Teoriya Veroyatnostei i ee Primeneniya, 1978, Volume 23, Issue 2, Pages 295–312
(Mi tvp3038)
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This article is cited in 15 scientific papers (total in 15 papers)
On the central limit theorem for Markov chains
В. A. Lifšic Leningrad
Abstract:
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 $\beta_n=1$
($n=1,2,\dots$).
Received: 15.12.1975
Citation:
В. A. Lifšic, “On the central limit theorem for Markov chains”, Teor. Veroyatnost. i Primenen., 23:2 (1978), 295–312; Theory Probab. Appl., 23:2 (1979), 279–296
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https://www.mathnet.ru/eng/tvp3038 https://www.mathnet.ru/eng/tvp/v23/i2/p295
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