Probability Theory and Related Fields

, Volume 84, Issue 2, pp 203–229

# Reinforced random walk

• Burgess Davis
Article

## Summary

Letai,i≧1, be a sequence of nonnegative numbers. Difine a nearest neighbor random motion$$\overrightarrow X$$=X0,X1, ... 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$$1 + \sum\limits_{j = 1}^k {a_j }$$. Given (X0,X1, ...,Xn)=(i0, i1, ..., in), the probability thatXn+1 isin−1 orin+1 is proportional to the weights at timen of the intervals (in−1,in) and (in,iin+1). We prove that$$\overrightarrow X$$ 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$$\mathop {\lim }\limits_{n \to \infty }$$X n /n=0 a.s. For much more general reinforcement schemes we proveP ($$\overrightarrow X$$ visits all integers infinitely often)+P ($$\overrightarrow X$$ has finite range)=1.

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