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, Acta Math. 131, 207-248) that X-t has a distribution with regularly varying tails of index alpha > 0 provided the equation E\a + bZ(1)\(u) = 1 has the solution u = alpha. 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 alpha <4. If alpha is an element of (0,2) we prove that the sample autocorrelations converge to non-degenerate limits. If alpha is an element of (2,4) we prove joint weak convergence of the sample autocorrelations and autocovariances to non-normal limits. (C) 1999 Elsevier Science B.V. All rights reserved.