The asymptotic behavior of quadratic forms in heavy-tailed strongly dependent random variables

AbstractSuppose that Xt = ∑∞j=0cjZt−j is a stationary linear sequence with regularly varying cj's and with innovations {Zj} that have infinite variance. Such a sequence can exhibit both high variability and strong dependence. The quadratic form Qn = ∑nt1s=1\̂gh(t − s)XtXs plays an important role in the estimation of the intensity of strong dependence. In contrast with the finite variance case, n−12(Qn − EQn) does not converge to a Gaussian distribution. We provide conditions on the cj's and on \̂gh for the quadratic form Qn, adequately normalized and randomly centered, to converge to a stable law of index α, 1 < α < 2, as n tends to infinity