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The range of a simple random walk on ℤ

Part of: Markov processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

P. Vallois
P. Vallois*
Affiliation:
Université de Nancy I
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Abstract

Let θ (a) be the first time when the range (Rn; n ≧ 0) is equal to a, Rn being equal to the difference of the maximum and the minimum, taken at time n, of a simple random walk on ℤ. We compute the g.f. of θ (a); this allows us to compute the distributions of θ (a) and Rn. We also investigate the asymptotic behaviour of θ (n), n going to infinity.

Keywords

STOPPING TIMERANDOM WALK

MSC classification

Primary: 60F05: Central limit and other weak theorems 60G40: Stopping times; optimal stopping problems; gambling theory 60G42: Martingales with discrete parameter 60G50: Sums of independent random variables; random walks
Secondary: 60J65: Brownian motion
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 1014 - 1033
Copyright
Copyright © Applied Probability Trust 1996 

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