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|d−αL(|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.
