On the lower tail probabilities of some random series

Lifshits, M. A.

Ann. Probab., Tome 25 (1997) no. 4, p. 424-442 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1997-01-14
Classification:  Small balls,  lower tail probabilities,  Laplace transform,  central limit theorem,  sums of independent variables,  60F10,  60G15