# Reinforced random walk

- Received:
- Revised:

DOI: 10.1007/BF01197845

- Cite this article as:
- Davis, B. Probab. Th. Rel. Fields (1990) 84: 203. doi:10.1007/BF01197845

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## Summary

Let*a*_{i},*i*≧1, be a sequence of nonnegative numbers. Difine a nearest neighbor random motion\(\overrightarrow X \)=*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 time*n* an interval (*i, i+1*) has been crossed exactly*k* times by the motion, its weight is\(1 + \sum\limits_{j = 1}^k {a_j } \). Given (*X*_{0},*X*_{1}, ...,*X*_{n})=(i_{0}, i_{1}, ..., i_{n}), the probability that*X*_{n+1} is*i*_{n}−1 or*i*_{n}+1 is proportional to the weights at time*n* of the intervals (*i*_{n}−1,*i*_{n}) and (*i*_{n},*i*i_{n}+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 prove*P* (\(\overrightarrow X \) visits all integers infinitely often)+*P* (\(\overrightarrow X \) has finite range)=1.