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Teoriya Veroyatnostei i ee Primeneniya, 1976, Volume 21, Issue 1, Pages 135–142 (Mi tvp3281)  

This article is cited in 4 scientific papers (total in 4 papers)

Short Communications

Local limit theorems for weighted sums of independent random variables

E. M. Shoukry

Leningrad
Full-text PDF (402 kB) Citations (4)
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.
Received: 17.09.1974
English version:
Theory of Probability and its Applications, 1976, Volume 21, Issue 1, Pages 137–144
DOI: https://doi.org/10.1137/1121011
Bibliographic databases:
Document Type: Article
Language: Russian
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://mathscinet.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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  • https://www.mathnet.ru/eng/tvp/v21/i1/p135
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Теория вероятностей и ее применения Theory of Probability and its Applications
     
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