The Sample ACF of a Simple Bilinear Process

We consider a simple bilinear process X t = aX t?1 +bX t?1 Z t?1 +Z t , where (Z t) is a sequence of iid N(0; 1) random variables. It follows from a result by Kesten (1973) that X t has a distribution with regularly varying tails of index > 0 provided the equation Eja + bZ 1 j u = 1 has the solution u =. We study the limit behaviour of the sample autocorrelations and autocovariances of this heavy{tailed non{linear process. Of particular interest is the case when < 4. If 2 (0; 2) we prove that the sample autocorrelations converge to non{degenerate limits. If 2 (2; 4) we prove joint weak convergence of the sample autocorrelations and autocovariances to an innnite variance =2{stable limit vector.