The sample ACF of a simple bilinear process

Abstract

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 α>0 provided the equation E| a+ bZ 1| u =1 has the solution u= α. 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 α<4. If α∈(0,2) we prove that the sample autocorrelations converge to non-degenerate limits. If α∈(2,4) we prove joint weak convergence of the sample autocorrelations and autocovariances to non-normal limits.