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January, 1987 Lower Tail Probability Estimates for Subordinators and Nondecreasing Random Walks
Naresh C. Jain, William E. Pruitt
Ann. Probab. 15(1): 75-101 (January, 1987). DOI: 10.1214/aop/1176992257


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 $t \rightarrow 0$.


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Naresh C. Jain. William E. Pruitt. "Lower Tail Probability Estimates for Subordinators and Nondecreasing Random Walks." Ann. Probab. 15 (1) 75 - 101, January, 1987.


Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0617.60023
MathSciNet: MR877591
Digital Object Identifier: 10.1214/aop/1176992257

Primary: 60F10
Secondary: 60G50

Keywords: local limit theorem , lower tail , nonnegative summands , Random walk , stochastic compactness , Subordinators

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 1 • January, 1987
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