Reinforced random walk TeX Export

@article{citeulike:2218982, abstract = {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.}, author = {Davis, Burgess}, citeulike-article-id = {2218982}, citeulike-linkout-0 = {http://dx.doi.org/10.1007/BF01197845}, day = {1}, doi = {10.1007/BF01197845}, journal = {Probability Theory and Related Fields}, keywords = {random, simulation, walk}, month = {June}, number = {2}, pages = {203--229}, posted-at = {2008-01-15 01:52:33}, priority = {5}, title = {Reinforced random walk}, url = {http://dx.doi.org/10.1007/BF01197845}, volume = {84}, year = {1990} }

TY - JOUR ID - citeulike:2218982 L3 - citeulike-article-id:2218982 N2 - 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. IS - 2 JF - Probability Theory and Related Fields EP - 229 TI - Reinforced random walk VL - 84 SP - 203 KW - random KW - simulation KW - walk AU - Davis, Burgess PY - 1990/06/01/ UR - http://dx.doi.org/10.1007/BF01197845 ER -