Limit theorems for a~random walk of a~special kind
Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 560-566
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Let $X_0\equiv0$, $X_1,\dots,X_n,\dots,$ be a Markov chain with the transition probabilities
\begin{gather*}
\mathbf P\{X_{n+1}=m+1\mid X_n=m\}=p(n,m),
\\
\mathbf P\{X_{n+1}=m\mid X_n=m\}=1-p(n,m).
\end{gather*} Recurrent relations are derived for the characteristic functions of the random variables $X_n$. On this basis for the cases $p(n,m)=\alpha+\varphi(n)$ and $p(n,m)=(n-m)/n$ Gärding's integral theorem (about the convergence of the appropriately normed and centered random variables $X_n$ to a normal random variable) is precised and a local limit theorem with an estimation of the speed of the convergence is proved
@article{TVP_1965_10_3_a18,
author = {S. G. Maloshevskii},
title = {Limit theorems for a~random walk of a~special kind},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {560--566},
publisher = {mathdoc},
volume = {10},
number = {3},
year = {1965},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1965_10_3_a18/}
}
S. G. Maloshevskii. Limit theorems for a~random walk of a~special kind. Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 560-566. http://geodesic.mathdoc.fr/item/TVP_1965_10_3_a18/