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Teor. Veroyatnost. i Primenen., 1976, Volume 21, Issue 1, Pages 135–142 (Mi tvp3281)  

Short Communications

Local limit theorems for weighted sums of independent random variables

E. M. Shoukry

Leningrad

Abstract: In this paper, we study the behaviour of $\displaystyle S_n=\sum_{k=-\infty}^{\infty}a_{kn}\xi_k$ as $n$ tends to infinity, where $\xi_k$ are independent identically distributed random variables and their common distribution function belongs to the domain of attraction of a certain stable law $G$ with index $\alpha$. Let the following two conditions on the matrix of coefficients ($a_{kn}$) be satisfied:
1) $\displaystyle\sum_{k=-\infty}^{\infty}|a_{kn}|^{\alpha}\widetilde h(a_{kn})=b_n\to 1\qquad(n\to\infty),
$ where $\widetilde h(x)$ is the slowly varying function from the representation for the characteristic function of $G$;
2) $\displaystyle\gamma_n=\sup_k|a_{kn}|\to 0\qquad(n\to\infty).
$ Then it is shown that the distribution function of $S_n$ converges to a stable distribution function, and, if $\displaystyle \int_{-\infty}^{\infty}|f(t)|^p dt<\infty$, $p>0$, where $f(t)$ is the characteristic function of $\xi_k$ then the density function of $S_n$ exists and converges to the density function of the limit distribution.

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English version:
Theory of Probability and its Applications, 1976, 21:1, 137–144

Bibliographic databases:

Received: 17.09.1974

Citation: E. M. Shoukry, “Local limit theorems for weighted sums of independent random variables”, Teor. Veroyatnost. i Primenen., 21:1 (1976), 135–142; Theory Probab. Appl., 21:1 (1976), 137–144

Citation in format AMSBIB
\Bibitem{Shu76}
\by E.~M.~Shoukry
\paper Local limit theorems for weighted sums of independent random variables
\jour Teor. Veroyatnost. i Primenen.
\yr 1976
\vol 21
\issue 1
\pages 135--142
\mathnet{http://mi.mathnet.ru/tvp3281}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=420796}
\zmath{https://zbmath.org/?q=an:0368.60061}
\transl
\jour Theory Probab. Appl.
\yr 1976
\vol 21
\issue 1
\pages 137--144
\crossref{https://doi.org/10.1137/1121011}


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