PARAMETER-ESTIMATION FOR ARMA MODELS WITH INFINITE VARIANCE INNOVATIONS

We consider a standard ARMA process of the form phi(B)X(t) = B(B)Z(t), where the innovations Z(t) belong to the domain of attraction of a stable law, so that neither the Z(t) nor the X(t) have a finite variance. Our aim is to estimate the coefficients of phi and theta. Since maximum likelihood estimation is not a viable possibility (due to the unknown form of the marginal density of the innovation sequence), we adopt the so-called Whittle estimator, based on the sample periodogram of the X sequence. Despite the fact that the periodogram does not, a priori, seem like a logical object to study in this non-L(2) situation, we show that our estimators are consistent, obtain their asymptotic distributions and show that they converge to the true values faster than in the usual L(2) case

The expected number of level crossings for stationary, harmonisable, symmetric, stable processes

This paper to appear in "Stochastic Processes and Their Applications"