The Annals of Probability

On the lower tail probabilities of some random series

M. A. Lifshits

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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

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

First available in Project Euclid: 18 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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