Some exact distributions of the number of one-sided deviations and the time of the last such deviation in the simple random walk
type: Item
instance of: scholarly article
title: Some exact distributions of the number of one-sided deviations and the time of the last such deviation in the simple random walk
zbMATH Open document ID: 0623.60046
DOI: 10.1016/0304-4149(87)90018-4
author: Chern-Ching Chao
author: John Slivka
published in: Stochastic Processes and their Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \(S_ n\) be the n-th partial sum of i.i.d. Bernoulli r.v. with probabilities p \((0<p<1)\) and \(q=1-p\) for taking values \(+1\) and -1, respectively, and with mean \(\mu =p-q\). For a fixed positive real number \(\lambda\), let \(N^+ (N^*)\) be the total number of values of n for which \(S_ n>(\lambda +\mu)n (S_ n\geq (\lambda +\mu)n)\) and let \(L^+ (L^*)\) be the supremum of the values of n for which \(S_ n>(\lambda +\mu)n (S_ n\geq (\lambda +\mu)n)\). Explicit expressions for the exact distributions of \(N^+\), \(N^*\), \(L^+\), \(L^*\) are given in this paper when \(\lambda +\mu =\pm k/(k+2)\) for any nonnegative integer k.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 4011556
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: Bernoulli variables
zbMATH Keywords: linear boundary crossings
zbMATH Keywords: last exit time
zbMATH Keywords: exact distributions
reviewed by: Chun Su
MaRDI profile type: Publication
full work available at URL: 0304-4149(87)90018-4
OpenAlex ID: W2039635383
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description: scientific article; zbMATH DE number 4011556