Reinforced Random Walk

Abstract

Leta i,i≧1, be a sequence of nonnegative numbers. Difine a nearest neighbor random motion X→\overrightarrow X =X 0,X 1, ... on the integers as follows. Initially the weight of each interval (i, i+1), i an integer, equals 1. If at timen an interval (i, i+1) has been crossed exactlyk times by the motion, its weight is 1+∑j=1kaj1 + \sum\limits_{j = 1}^k {a_j } . Given (X 0,X 1, ...,X n)=(i0, i1, ..., in), the probability thatX n+1 isi n−1 ori n+1 is proportional to the weights at timen of the intervals (i n−1,i n) and (i n,iin+1). We prove that X→\overrightarrow X either visits all integers infinitely often a.s. or visits a finite number of integers, eventually oscillating between two adjacent integers, a.s., and that limn→∞\mathop {\lim }\limits_{n \to \infty } X n /n=0 a.s. For much more general reinforcement schemes we proveP ( X→\overrightarrow X visits all integers infinitely often)+P ( X→\overrightarrow X has finite range)=1.