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 α-stable distribution with 1 < α < 2, and let cj = j−βΛ(j) with 1/α < β and a slowly varying Λ(j). We study the limit behavior of the partial sum processes Dn(t) = ∑[nt]k=1Q(∑k−∞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.