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Teoriya Veroyatnostei i ee Primeneniya, 1965, Volume 10, Issue 3, Pages 560–566
(Mi tvp556)
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This article is cited in 1 scientific paper (total in 1 paper)
Short Communications
Limit theorems for a random walk of a special kind
S. G. Maloshevskii
Abstract:
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
Received: 12.12.1964
Citation:
S. G. Maloshevskii, “Limit theorems for a random walk of a special kind”, Teor. Veroyatnost. i Primenen., 10:3 (1965), 560–566; Theory Probab. Appl., 10:3 (1965), 507–512
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