Simple random walk - Uppsala University

Simple random walkSven Erick Alm9 April 2002(revised 8 March 2006)(translated to English 28 March 2006)Contents1 Introduction22 The monkey at the Passage probabilities .. Passage times .. 53 The gambler s Absorption probabilities .. Absorption times .. Reflecting barriers .. 104 Counting Mirroring .. The Ballot problem .. Recurrence .. Maximum .. The Arcsine law .. 205 Mixed problems236 Literature2411 IntroductionArandom walkis a stochastic sequence{Sn}, withS0= 0, defined bySn=n k=1Xk,where{Xk}are independent and identically distributed random variables ( ).The random walk issimpleifXk= 1, withP(Xk= 1) =pandP(Xk= 1) = 1 p=q. Imagine a particle performing a random walk on the integer points of the real line, where itin each step moves to one of its neighboring points; see Figure 1: Simple random walkRemark can also study random walks in higher dimensions.

The monkey at the cliff can be interpreted as placing an absorbing barrier at x = 1 (or x = k). By studying a random walk with two absorbing barriers, one on each side of the staring point, we can solve The Gambler’s ruin: Two players, A and B, play a game with independent rounds where, in each round, one

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