Let $X_1, X_2,\ldots$ be nonnegative i.i.d. random variables and $S_n = X_1 + \cdots + X_n; EX_1 = \mu \leq \infty$ and $a$ is the infimum of the support of the distribution of $X_1$. For $a < x_n < \mu$ we obtain the asymptotic behavior of $\log P\{S_n \leq nx_n\}$ as $n \rightarrow \infty$. Under the additional assumption of stochastic compactness a stronger result is obtained which gives the asymptotic behavior of $P\{S_n \leq nx_n\}$ itself. Analogues of these results are given for subordinators when $t \rightarrow \infty$ or $t \rightarrow 0$.
Publié le : 1987-01-14
Classification:
Random walk,
nonnegative summands,
lower tail,
stochastic compactness,
subordinators,
local limit theorem,
60F10,
60G50
@article{1176992257,
author = {Jain, Naresh C. and Pruitt, William E.},
title = {Lower Tail Probability Estimates for Subordinators and Nondecreasing Random Walks},
journal = {Ann. Probab.},
volume = {15},
number = {4},
year = {1987},
pages = { 75-101},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992257}
}
Jain, Naresh C.; Pruitt, William E. Lower Tail Probability Estimates for Subordinators and Nondecreasing Random Walks. Ann. Probab., Tome 15 (1987) no. 4, pp. 75-101. http://gdmltest.u-ga.fr/item/1176992257/