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Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 3, Pages 424–438 (Mi tvp2655)  

This article is cited in 1 scientific paper (total in 1 paper)

On the limiting distribution classes for sums of a random number of independent identically distributed random variables

D. Szász

Budapest

Abstract: Let, for every $n$, $\{\xi_{nk}\}_{k=1,…,k_n}$ be independent identically distributed random variables and $\nu_n$ be a non-negative integer-valued random variable independent of the random variables $\{\xi_{nk}\}_{k=1,…,k_n}$. Put $S_k^{(n)}=\xi_{n1}+…+\xi_{nk}$ and suppose that $\nu_n\to\infty$ in probability. It is proved that, if
$$ \mathbf P\{S_{\nu_n}^{(n)}<x\}\to\Phi(x), $$
where $\Phi(x)$ is an arbitrary distribution function, then there exists a sequence of integers $k_n$ such that the sequences of random variables $\{S_{k_n}^{(n)}\}_{n=1,2,…$ and $\{\nu_n/k_n\}_{n=1,2,…$are weakly compact. On the basis of this fact, limiting classes in (1) are characterized and necessary conditions of the convergence (1) are given.

Full text: PDF file (730 kB)

English version:
Theory of Probability and its Applications, 1973, 17:3, 401–415

Bibliographic databases:

Received: 24.11.1970

Citation: D. Szász, “On the limiting distribution classes for sums of a random number of independent identically distributed random variables”, Teor. Veroyatnost. i Primenen., 17:3 (1972), 424–438; Theory Probab. Appl., 17:3 (1973), 401–415

Citation in format AMSBIB
\Bibitem{Sza72}
\by D.~Sz\'asz
\paper On the limiting distribution classes for sums of a~random number of independent identically distributed random variables
\jour Teor. Veroyatnost. i Primenen.
\yr 1972
\vol 17
\issue 3
\pages 424--438
\mathnet{http://mi.mathnet.ru/tvp2655}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=307304}
\zmath{https://zbmath.org/?q=an:0277.60016}
\transl
\jour Theory Probab. Appl.
\yr 1973
\vol 17
\issue 3
\pages 401--415
\crossref{https://doi.org/10.1137/1117050}


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    This publication is cited in the following articles:
    1. Theory Probab. Appl., 58:1 (2014), 17–24  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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