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January 1997 On the lower tail probabilities of some random series
M. A. Lifshits
Ann. Probab. 25(1): 424-442 (January 1997). DOI: 10.1214/aop/1024404294


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.


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M. A. Lifshits. "On the lower tail probabilities of some random series." Ann. Probab. 25 (1) 424 - 442, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0873.60012
MathSciNet: MR1428515
Digital Object Identifier: 10.1214/aop/1024404294

Primary: 60F10
Secondary: 60G15

Keywords: central limit theorem , Laplace transform , lower tail probabilities , Small balls , sums of independent variables

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 1 • January 1997
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