2009/11/19 by Olivier Raimond, Bruno Schapira, Raimond, Olivier +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.0911.3886
previous Section 2 removed, to appear in Illinois J. Math
arxiv created 2012/04/11 · arxiv updated 2012/04/12
We study recurrence properties and the validity of the (weak) law of large numbers for (discrete time) processes which, in the simplest case, are obtained from simple symmetric random walk on \Z by modifying the distribution of a step from a fresh point. If the process is denoted as \Sn\n ≥ 0, then the conditional distribution of Sn+1 - Sn given the past through time n is the distribution of a simple random walk step, provided Sn is at a point which has been visited already at least once during [0,n-1]. Thus in this case P\Sn+1-Sn = ± 1|S_ℓ, ℓ ≤ n\ = 1/2. We denote this distribution by P1. However, if Sn is at a point which has not been visited before time n, then we take for the conditional distribution of Sn+1-Sn, given the past, some other distribution P2. We want to decide in specific cases whether Sn returns infinitely often to the origin and whether (1/n)Sn → 0 in probability. Generalizations or variants of the Pi and the rules for switching between the Pi are also considered.