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April, 1974 On the Moments and Limit Distributions of Some First Passage Times
Allan Gut
Ann. Probab. 2(2): 277-308 (April, 1974). DOI: 10.1214/aop/1176996709


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.


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Allan Gut. "On the Moments and Limit Distributions of Some First Passage Times." Ann. Probab. 2 (2) 277 - 308, April, 1974.


Published: April, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0278.60031
MathSciNet: MR394857
Digital Object Identifier: 10.1214/aop/1176996709

Primary: 60G40
Secondary: 60F05 , 60G45 , 60G50 , 60K05

Keywords: continuous from above , excess over the boundary , extended renewal theory , First passage time , ladder height , ladder index , regular variation , renewal theory , separable random process , slow variation , stopping time

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 2 • April, 1974
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