Random walks and Riesz kernels
John H. Williamson · Pacific Journal of Mathematics · 1968
It is the purpose of this paper to study the behavior for large \x -y\ of the Green Function, G(x, y\ of a random walk, {S n , n e N} 9 having increments belonging to the domain of attraction of a ^-dimensional stable law with characteristic exponent a, 0 < a < min (d, 2).The main results are concerned with the problem of finding conditions under which G(0, x) is asymptotic to | x \ d -«L(\ x |) where L is a function of slow growth.The results, including those found in the discussion of the discrete potential theory for such random walks, are for the most part discrete analogs of theorems for transient stable processes.Throughout this paper the notation, definitions, and conventions of [16] will be used.The position of the random walk at time n will be denoted by S n with S Q = 0.The independent identically distributed random variables, S k -S fc _i, will be denoted by X k .P will refer to the measure on the underlying probability space.P{dω} .P n (x, y) = P n (0, y-x) = P{S n = y -x], n = 1, 2, ... .P(x, V) = Pi(x, V) and δ(x, y) = P 0 (x, y) .