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

• http://mi.mathnet.ru/eng/tvp2655
• http://mi.mathnet.ru/eng/tvp/v17/i3/p424

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