Geodesic
Necessary and sufficient conditions for weak convergence of random sums of independent random variables
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 465-482.

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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).
Classification : 60F05, 60G50
Keywords: random sums; weak convergence; stable law; nonrandom centering; measure of dependence between $\sigma $-fields 
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     author = {Krajka, A. and Rychlik, Z.},
     title = {Necessary and sufficient conditions for weak convergence of random sums of independent random variables},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {465--482},
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     zbl = {0785.60016},
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Krajka, A.; Rychlik, Z. Necessary and sufficient conditions for weak convergence of random sums of independent random variables. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 465-482. http://geodesic.mathdoc.fr/item/CMUC_1993__34_3_a9/