Necessary and sufficient conditions for weak convergence of random sums of independent random variables
Krajka, Andrzej ; Rychlik, Zdzisław
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993), p. 465-482 / Harvested from Czech Digital Mathematics Library
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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).

Publié le : 1993-01-01
Classification:  60F05,  60G50 
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     author = {Andrzej Krajka and Zdzis\l aw Rychlik},
     title = {Necessary and sufficient conditions for weak convergence of random sums of independent random variables},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {34},
     year = {1993},
     pages = {465-482},
     zbl = {0785.60016},
     mrnumber = {1243079},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118604}
}

Krajka, Andrzej; Rychlik, Zdzisław. Necessary and sufficient conditions for weak convergence of random sums of independent random variables. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 465-482. http://gdmltest.u-ga.fr/item/118604/

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