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February, 1978 Probability Bounds for First Exits Through Moving Boundaries
Stephen Portnoy
Ann. Probab. 6(1): 106-117 (February, 1978). DOI: 10.1214/aop/1176995614


Let $S_1, S_2,\cdots$ be partial sums of independent and identically distributed random variables and let $f(n)$ and $g(n)$ be increasing positive sequences. Nearly sharp bounds are presented for the probabilities $P\{S_i \geqq g(i), i = 1,\cdots, n\}$ and $P\{- f(i) \leqq S_i \leqq f(i), i = 1,\cdots, n\}$ under conditions on $f$ and $g$. The most difficult results are the lower bounds in the normal case. Results are obtained by an embedding method which approximates Brownian motion by sums of independent random variables taking on only two or three values.


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Stephen Portnoy. "Probability Bounds for First Exits Through Moving Boundaries." Ann. Probab. 6 (1) 106 - 117, February, 1978.


Published: February, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0379.60066
MathSciNet: MR458601
Digital Object Identifier: 10.1214/aop/1176995614

Primary: 60J15
Secondary: 60G40 , 60G50 , 60J65

Keywords: First exit times , moving boundaries , Random walks

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 1 • February, 1978
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