Geodesic
Convergence of moments in the central limit theorem for nonstationary Markov chains
Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 4, pp. 755-771 
В. A. Lifšic. Convergence of moments in the central limit theorem for nonstationary Markov chains. Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 4, pp. 755-771. http://geodesic.mathdoc.fr/item/TVP_1975_20_4_a4/
@article{TVP_1975_20_4_a4,
     author = {{\CYRV}. A. Lif\v{s}ic},
     title = {Convergence of moments in the central limit theorem for nonstationary {Markov} chains},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {755--771},
     year = {1975},
     volume = {20},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1975_20_4_a4/}
}
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VL  - 20
IS  - 4
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ID  - TVP_1975_20_4_a4
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%D 1975
%P 755-771
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%G ru
%F TVP_1975_20_4_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

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} $N(0,1)$.