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On the rate of growth of the overshoot and the maximum partial sum

Part of: Stochastic processes Special processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

P. S. Griffin  and
R. A. Maller
P. S. Griffin*
Affiliation:
Syracuse University
R. A. Maller*
Affiliation:
University of Western Australia
*
∗ Postal address: Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150, USA.
∗∗ Postal address: Department of Mathematics, The University of Western Australia, Nedlands 6907, Western Australia.
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Abstract

Let Tr be the first time at which a random walk Sn escapes from the strip [-r,r], and let |STr|-r be the overshoot of the boundary of the strip. We investigate the order of magnitude of the overshoot, as r → ∞, by providing necessary and sufficient conditions for the ‘stability’ of |STr|, by which we mean that |STr|/r converges to 1, either in probability (weakly) or almost surely (strongly), as r → ∞. These also turn out to be equivalent to requiring only the boundedness of |STr|/r, rather than its convergence to 1, either in the weak or strong sense, as r → ∞. The almost sure characterisation turns out to be extremely simple to state and to apply: we have |STr|/r → 1 a.s. if and only if EX2 < ∞ and EX = 0 or 0 < |EX| ≤ E|X| < ∞. Proving this requires establishing the equivalence of the stability of STr with certain dominance properties of the maximum partial sum Sn* = max{|Sj|: 1 ≤ j ≤ n} over its maximal increment.

Keywords

Renewal theoremsrandom walksovershootChung lawexit time from a stripstability

MSC classification

Primary: 60K05: Renewal theory 60F15: Strong theorems 60F05: Central limit and other weak theorems
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 181 - 196
Copyright
Copyright © Applied Probability Trust 1998 

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