The behavior of tail probabilities $\mathbf{P}{S \leq r}, r \to 0$
is investigated, where $S$ is a series $S = \Sigma \lambda_j Z_j$ generated by
some sequence of positive numbers ${\lambda_j}$ and by a sequence ${Z_j}$ of
independent copies of a positive random variable $Z$.
¶ We present the exact asymptotic expression for $\mathbf{P}{S \leq
r}$ by means of Laplace transform $\Lambda (\gamma) = \mathbf{E} \exp {- \gamma
S}$ under weak assumptions on the behavior of the tail probabilities of $Z$ in
the vicinity of zero. The bounds of accuracy are also given, and under weak
supplementary smoothness conditions the asymptotic properties of the density of
$S$ are investigated.
Publié le : 1997-01-14
Classification:
Small balls,
lower tail probabilities,
Laplace transform,
central limit theorem,
sums of independent variables,
60F10,
60G15
@article{1024404294,
author = {Lifshits, M. A.},
title = {On the lower tail probabilities of some random series},
journal = {Ann. Probab.},
volume = {25},
number = {4},
year = {1997},
pages = { 424-442},
language = {en},
url = {http://dml.mathdoc.fr/item/1024404294}
}
Lifshits, M. A. On the lower tail probabilities of some random series. Ann. Probab., Tome 25 (1997) no. 4, pp. 424-442. http://gdmltest.u-ga.fr/item/1024404294/