科研之友
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摘要
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 {S-n}(n >= 0), then the conditional distribution of Sn+1 - S-n given the past through time n is the distribution of a simple random walk step, provided S-n is at a point which has been visited already at least once during [0, n - 1]. Thus, in this case, P{Sn+1 - S-n = +/- 1 vertical bar S-l, l <= n} = 1/2. We denote this distribution by P-1. However, if S-n is at a point which has not been visited before time n, then we take for the conditional distribution of Sn+1 - S-n, given the past, some other distribution P-2. We want to decide in specific cases whether S-n returns infinitely often to the origin and whether (1/n)S-n -> 0 in probability. Generalizations or variants of the P-i and the rules for switching between the P-i are also considered.
- 出版日期2010
