ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01197845
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