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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References
Burkholder, D.L.: Martingale transforms. Ann. Math. Statist.37, 1494–1504 (1966)
Burkholder, D.L., Gundy, R.F.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math.124, 249–304 (1970)
Doob, J.L.: Stochastic processes. New York: Wiley 1953
Dubins, L.E.: A note on upcrossings of semimartingales. Ann. Math. Statist.37, 728 (1966)
Karlin, S., Taylor, H.M.: A first course in stochastic processes, 2n edn. New York: Academic Press 1975
Luce, R.D.: Individual choice behavior: A theoretical analysis. New York: Wiley 1959
Neveu, J.: Mathematical foundations of the calculus of probability. San Francisco: Holden-Day 1965
Pemantle, R.: Phase transition in reinforced random walk and RWRE on trees. Ann. Probab.16, 1229–1241 (1988)
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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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DOI: https://doi.org/10.1007/BF01197845