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### A central limit theorem for sums of a random number of independent random variables

Colloquium Mathematicae

### An almost sure central limit theorem for independent random variables

Annales de l'I.H.P. Probabilités et statistiques

### The invariance principle for nonstationary sequences of associated random variables

Annales de l'I.H.P. Probabilités et statistiques

### Necessary and sufficient conditions for weak convergence of random sums of independent random variables

Commentationes Mathematicae Universitatis Carolinae

Let $\left\{{X}_{n},\phantom{\rule{0.166667em}{0ex}}n\ge 1\right\}$ be a sequence of independent random variables such that $E{X}_{n}={a}_{n}$, $E{\left({X}_{n}-{a}_{n}\right)}^{2}={\sigma }_{n}^{2}$, $n\ge 1$. Let $\left\{{N}_{n},\phantom{\rule{0.166667em}{0ex}}n\ge 1\right\}$ be a sequence od positive integer-valued random variables. Let us put ${S}_{{N}_{n}}={\sum }_{k=1}^{{N}_{n}}{X}_{k}$, ${L}_{n}={\sum }_{k=1}^{n}{a}_{k}$, ${s}_{n}^{2}={\sum }_{k=1}^{n}{\sigma }_{k}^{2}$, $n\ge 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\left\{\left({S}_{{N}_{n}}-{L}_{n}\right)/{s}_{n},\phantom{\rule{0.166667em}{0ex}}n\ge 1\right\}$, as $n\to \infty$. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).

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