## The Annals of Probability

### Lower Tail Probability Estimates for Subordinators and Nondecreasing Random Walks

#### Abstract

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 $t \rightarrow 0$.

#### Article information

Source
Ann. Probab., Volume 15, Number 1 (1987), 75-101.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992257

Digital Object Identifier
doi:10.1214/aop/1176992257

Mathematical Reviews number (MathSciNet)
MR877591

Zentralblatt MATH identifier
0617.60023

JSTOR