The sample ACF of a simple bilinear process

We consider a simple bilinear process Xt=aXt-1+bXt-1Zt-1+Zt, where (Zt) is a sequence of iid N(0,1) random variables. It follows from a result by Kesten (1973, Acta Math. 131, 207-248) that Xt has a distribution with regularly varying tails of index [alpha]>0 provided the equation Ea+bZ1u=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][set membership, variant](0,2) we prove that the sample autocorrelations converge to non-degenerate limits. If [alpha][set membership, variant](2,4) we prove joint weak convergence of the sample autocorrelations and autocovariances to non-normal limits.