Transcription of ONE-DIMENSIONAL RANDOM WALKS
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ONE-DIMENSIONAL RANDOM WALKS
ONE-DIMENSIONAL RANDOM WALKS1. SIM P LERA NDO MWA LKDefinition walkon the integersZwith step distributionFand initial statex Zis a sequenceSnof RANDOM variables whose increments are independent, identically distributedrandom variables iwith common distributionF, that is,(1)Sn=x+n i=1 definition extends in an obvious way to RANDOM WALKS on thed dimensional integer lat-ticeZd: the increments are then randomd RANDOM walkonZdis the particularcase where the step distribution is the uniform distribution on the 2dnearest neighbors of theorigin; in one dimension, this is theRademacher-12distribution, the distribution that puts mass1/2 at each of the two values 1.
post- y process is just an independent simple random walk started at y. But (10) (with the roles of x,y reversed) implies that this random walk must eventually visit x. When this happens, the random walk restarts again, so it will go back to y, and so on. Thus, by an easy induction argu-ment (see Corollary 14 below): Theorem 4.
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