Limit Behavior of Quadratic Forms of Moving Averages and Statistical Solutions of the Burgers' Equation

Let ..., X-1, X0, X1, ... be symmetric i.i.d. random variables belonging to the domain of normal attraction of an [alpha]-stable distribution with 1 < [alpha] < 2, and let cj = j-[beta][Lambda](j) with 1/[alpha] < [beta] and a slowly varying [Lambda](j). We study the limit behavior of the partial sum processes Dn(t) = [summation operator][nt]k=1Q([summation operator]k-[infinity]ck-j) Xj), where Q(x) is a secondorder polynomial. We also comment on how our result relates to the study of the Burgers' equation with random initial data.