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random sums; weak convergence; stable law; nonrandom centering; measure of dependence between $\sigma $-fields
Let $\{X_n,\, n\geq 1\}$ be a sequence of independent random variables such that $EX_n=a_n$, $E(X_n-a_n)^2=\sigma _n^2$, $n\geq 1$. Let $\{N_n,\, n\geq 1\}$ be a sequence od positive integer-valued random variables. Let us put $S_{N_n}=\sum_{k=1}^{N_n} X_k$, $L_n=\sum_{k=1}^{n} a_k$, $s_n^2=\sum_{k=1}^{n} \sigma _k^2$, $n\geq 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\{(S_{N_n}-L_n)/s_n,\, n\geq 1\}$, as $n\rightarrow \infty $. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).
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