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The exponential rate of convergence of the distribution of the maximum of a random walk

Published online by Cambridge University Press:  14 July 2016

N. Veraverbeke  and
J. L. Teugels
N. Veraverbeke
Affiliation:
Catholic University of Louvain
J. L. Teugels
Affiliation:
Catholic University of Louvain
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Abstract

Let Gn(x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn(x) — G(x) is asymptotically equal to c.H(x)n−3/2γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

Keywords

COMPLETE EXPONENTIAL CONVERGENCEMAXIMUM OF SUMS OF INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLESSUPREMUM OF PROCESSES WITH STATIONARY INDEPENDENT INCREMENTSWAITING TIME DISTRIBUTIONEXPONENTIAL RATE OF DECAY
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 2 , June 1975 , pp. 279 - 288
Copyright
Copyright © Applied Probability Trust 1975 

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