Vol. 25, No. 2, 1968

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Random walks and Riesz kernels

John Alexander Williamson

Vol. 25 (1968), No. 2, 393–415

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, {Sn,n N}, having increments belonging to the domain of attraction of a d-dimensional stable law with characteristic exponent α, 0 < α < min(d,2). The main results are concerned with the problem of finding conditions under which G(0,x) is asymptotic to |x|dαL(|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.

Mathematical Subject Classification
Primary: 60.66
Received: 28 April 1967
Published: 1 May 1968
John Alexander Williamson