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

D. Szász

Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 3, pp. 424-438

Résumé

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