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Reinforced random walk
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  • Published: June 1990

Reinforced random walk

  • Burgess Davis1 

Probability Theory and Related Fields volume 84, pages 203–229 (1990)Cite this article

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Summary

Leta 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 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 (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\(\overrightarrow X \) 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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Authors and Affiliations

  1. Department of Statistics, Purdue University, 47907, West Lafayette, IN, USA

    Burgess Davis

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  1. Burgess Davis
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Supported by a National Science Foundation Grant

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Davis, B. Reinforced random walk. Probab. Th. Rel. Fields 84, 203–229 (1990). https://doi.org/10.1007/BF01197845

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  • Received: 05 July 1988

  • Revised: 11 July 1989

  • Issue Date: June 1990

  • DOI: https://doi.org/10.1007/BF01197845

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Keywords

  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Finite Number
  • Mathematical Biology
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