The Annals of Probability

On the lower tail probabilities of some random series

M. A. Lifshits

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Abstract

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.

Article information

Source
Ann. Probab., Volume 25, Number 1 (1997), 424-442.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404294

Digital Object Identifier
doi:10.1214/aop/1024404294

Mathematical Reviews number (MathSciNet)
MR1428515

Zentralblatt MATH identifier
0873.60012

Subjects
Primary: 60F10: Large deviations
Secondary: 60G15: Gaussian processes

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

Citation

Lifshits, M. A. On the lower tail probabilities of some random series. Ann. Probab. 25 (1997), no. 1, 424--442. doi:10.1214/aop/1024404294. https://projecteuclid.org/euclid.aop/1024404294


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  • KOMENDANTSKII PROSPECT, 22-2-49 ST. PETERSBURG 197372 RUSSIA E-MAIL: mikhail@lifshits.spb.su