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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.

Gambler’s Ruin. Simple random walk describes (among other things) the fluctuations in a speculator’s wealth when he/she is fully invested in a risky asset whose value jumps by either 1 in each time period. Although this seems far too simple a model to be of any practical value,

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