Let $S_n, n = 1,2,3, \cdots$, denote the partial sums of i.i.d. random variables with positive, finite mean. The first passage times $\min \{n; S_n > c\}$ and $\min \{n; S_n > c \cdot a(n)\}$, where $c \geqq 0$ and $a(y)$ is a positive, continuous function on $\lbrack 0, \infty)$, such that $a(y) = o(y)$ as $y \uparrow \infty$, are investigated. Necessary and sufficient conditions for finiteness of their moments and moment generating functions are given. Under some further assumptions on $a(y)$, asymptotic expressions for the moments and the excess over the boundary are obtained when $c \rightarrow \infty$. Convergence to the normal and stable distributions is established when $c \rightarrow \infty$. Finally, some of the results are generalized to a class of random processes.
Publié le : 1974-04-14
Classification:
First passage time,
stopping time,
renewal theory,
extended renewal theory,
excess over the boundary,
ladder index,
ladder height,
regular variation,
slow variation,
separable random process,
continuous from above,
60G40,
60G50,
60F05,
60G45,
60K05
@article{1176996709,
author = {Gut, Allan},
title = {On the Moments and Limit Distributions of Some First Passage Times},
journal = {Ann. Probab.},
volume = {2},
number = {6},
year = {1974},
pages = { 277-308},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996709}
}
Gut, Allan. On the Moments and Limit Distributions of Some First Passage Times. Ann. Probab., Tome 2 (1974) no. 6, pp. 277-308. http://gdmltest.u-ga.fr/item/1176996709/