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Teoriya Veroyatnostei i ee Primeneniya, 1975, Volume 20, Issue 4, Pages 755–771 (Mi tvp3339)  

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, Volume 20, Issue 4, Pages 741–758
DOI: https://doi.org/10.1137/1120082
Bibliographic databases:
Language: Russian
Citation: В. A. Lifšic, “Convergence of moments in the central limit theorem for nonstationary Markov chains”, Teor. Veroyatnost. i Primenen., 20:4 (1975), 755–771; Theory Probab. Appl., 20:4 (1976), 741–758
Citation in format AMSBIB
\Bibitem{Lif75}
\by В.~A.~Lif{\v s}ic
\paper Convergence of moments in the central limit theorem for nonstationary Markov chains
\jour Teor. Veroyatnost. i Primenen.
\yr 1975
\vol 20
\issue 4
\pages 755--771
\mathnet{http://mi.mathnet.ru/tvp3339}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=423534}
\zmath{https://zbmath.org/?q=an:0426.60021}
\transl
\jour Theory Probab. Appl.
\yr 1976
\vol 20
\issue 4
\pages 741--758
\crossref{https://doi.org/10.1137/1120082}
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