IJPAM: Volume 88, No. 3 (2013)

AVERAGE EXTREMA OF A RANDOM WALK WITH
A NEGATIVE BINOMIAL STOPPING TIME

David K. Neal
Department of Mathematics
Western Kentucky University
Bowling Green, KY 42101, USA


Abstract. A random walk $X$ moves upward or downward one unit at a time with probabilities $p$ and $q = 1 - p$, respectively. We derive the averages of the maximum height and the minimum height attained before $n$ downward movements occur, and show the relationship between these average extrema and the average final height. We derive the limits of these average extrema as $n$ increases to $\infty$, and derive the condition on $p$ that makes the average extrema symmetric about the initial height of $X$.

Received: June 28, 2013

AMS Subject Classification: 60G50

Key Words and Phrases: simple random walk, average maximum height, negative binomial random variable, reflection principle

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DOI: 10.12732/ijpam.v88i3.7 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 88
Issue: 3