Weak compactness of random sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 43 (1998) no. 2, pp. 248-271
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The shift-compactness of random sums $S_{N_n}^{(n)}$, $S_k^{(n)}=X_{n,1}+\cdots +X_{n,k}$, of independent random variables is investigated under the assumptions that in each sum the summands and their number $N_n$ are independent and that the summands satisfy the condition of uniform asymptotic negligibility in the form $$ \max_{1\le k\le N_n}\mathsf{P}\{|X_{n,k}|\ge\varepsilon\}\to0 $$
in probability for each $\varepsilon>0$. Some necessary and sufficient conditions are given for the weak compactness of random sums $S_{N_n}^{(n)}-A_n$, and the form of centering constants $A_n$ is described.
Keywords:
random variable, distribution function, weak convergence, weak compactness, shift-compactness, random sum.
@article{TVP_1998_43_2_a2,
author = {V. M. Kruglov},
title = {Weak compactness of random sums of independent random variables},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {248--271},
publisher = {mathdoc},
volume = {43},
number = {2},
year = {1998},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1998_43_2_a2/}
}
V. M. Kruglov. Weak compactness of random sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 43 (1998) no. 2, pp. 248-271. http://geodesic.mathdoc.fr/item/TVP_1998_43_2_a2/