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\(\mathop {\lim }\limits_{n \to \infty } \)Xn/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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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Burgess Davis
    • 1
  1. 1.Department of StatisticsPurdue UniversityWest LafayetteUSA

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