## The Annals of Probability

### On the lower tail probabilities of some random series

M. A. Lifshits

#### 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

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

#### 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