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Unformatted text preview: Simple random walk Sven Erick Alm 9 April 2002 (revised 8 March 2006) (translated to English 28 March 2006) Contents 1 Introduction 2 2 The monkey at the cliff 3 2.1 Passage probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Passage times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 The gambler’s ruin 6 3.1 Absorption probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Absorption times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Reflecting barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Counting paths 11 4.1 Mirroring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 The Ballot problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.4 Maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.5 The Arcsine law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5 Mixed problems 23 6 Literature 24 1 1 Introduction A random walk is a stochastic sequence { S n } , with S = 0 , defined by S n = n X k =1 X k , where { X k } are independent and identically distributed random variables (i.i.d.). The random walk is simple if X k = ± 1 , with P ( X k = 1) = p and P ( X k = 1) = 1 p = q . Imagine a particle performing a random walk on the integer points of the real line, where it in each step moves to one of its neighboring points; see Figure 1. q p Figure 1: Simple random walk Remark 1. You can also study random walks in higher dimensions. In two dimensions, each point has 4 neighbors and in three dimensions there are 6 neighbors. A simple random walk is symmetric if the particle has the same probability for each of the neighbors. General random walks are treated in Chapter 7 in Ross’ book. Here we will only study simple random walks, mainly in one dimension. We are interested in answering the following questions: • What is the probability that the particle will ever reach the point a ? (The case a = 1 is often called “The monkey at the cliff”.) • What time does it take to reach a ? • What is the probability of reaching a > before b < ? (“The gambler’s ruin”) • If the particle after n steps is at a > , what is the probability that – it has been on the positive side since the first step? – it has never been on the negative side? (“The Ballot problem”) • How far away from 0 will the particle get in n steps? When analyzing random walks, one can use a number of general methods, such as • conditioning, • generating functions, • difference equations, 2 • the theory for Markov chains, • the theory for branching processes, • martingales, but also some more specialized, such as • counting paths, • mirroring, • time reversal....
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This note was uploaded on 01/21/2012 for the course STAT 333 taught by Professor Chisholm during the Fall '08 term at Waterloo.
 Fall '08
 Chisholm
 Probability

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