Some properties of random walk paths

AbstractFor random walk on the d-dimensional integer lattice we consider again the problem of deciding when a set is recurrent, that is visited infinitely often with probability one by the random walk in question. Some special cases are considered, among them the following: for d = 2, what sequences (nj) have the property that with probability one the random walk visits the origin for infinitely many nj. A related problem, which is however not a special case of the recurrence problem, is to decide for what sequences (nj) the states visited by the random walk at times nj are all distinct, with only a finite number of exceptions. This problem is dealt with in the final part of the paper