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Teor. Veroyatnost. i Primenen., 1965, Volume 10, Issue 3, Pages 560–566 (Mi tvp556)  

Short Communications

Limit theorems for a random walk of a special kind

S. G. Maloshevskii


Abstract: Let $X_0\equiv0$, $X_1,…,X_n,…,$ 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

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English version:
Theory of Probability and its Applications, 1965, 10:3, 507–512

Bibliographic databases:

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

Citation in format AMSBIB
\Bibitem{Mal65}
\by S.~G.~Maloshevskii
\paper Limit theorems for a~random walk of a~special kind
\jour Teor. Veroyatnost. i Primenen.
\yr 1965
\vol 10
\issue 3
\pages 560--566
\mathnet{http://mi.mathnet.ru/tvp556}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=190997}
\zmath{https://zbmath.org/?q=an:0209.19503}
\transl
\jour Theory Probab. Appl.
\yr 1965
\vol 10
\issue 3
\pages 507--512
\crossref{https://doi.org/10.1137/1110064}


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