Limit behavior of quadratic forms of moving averages and statistical solutions of the Burgers' equation (Q1343346)

Let \(\dots, X_{-1}, X_ 0, X_ 1, \dots\) 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 \(c_ j = j^{-\beta} \Lambda (j)\) with \(\alpha \beta > 1\) and slowly varying \(\Lambda (j)\). The limit behaviour of the partial sum processes \(D_ n(t) = \sum^{[nt]}_{k=1} Q(\sum^ k_{j=-\infty}c_{k-j} X_ j)\), where \(Q(x)\) is a second order polynomial, is studied. In the second part of the paper the problem of asymptotic behaviour of solutions of the Burgers' equation \(u_ t + uu_ x = \nu u_{xx}\), where \(\nu > 0\), with initial datum \(u(x,0) = u_ 0(x)\) being a strictly stationary process, is discussed. It is shown how the results of the first part can be applied here.