THE ASYMPTOTIC-BEHAVIOR OF QUADRATIC-FORMS IN HEAVY-TAILED STRONGLY DEPENDENT RANDOM-VARIABLES

Suppose that X(t) = Sigma(j = 0)(infinity)c(j)Z(t - j) is a stationary linear sequence with regularly varying c(j)'s and with innovations {Z (j)} that have infinite variance. Such a sequence can exhibit both hig h variability and strong dependence. The quadratic form Q(n) = Sigma(t 1s = 1)(n) <(eta)over cap>(t - s)X(t)X(s) plays an important role in t he estimation of the intensity of strong dependence. In contrast with the finite variance case, n(-1/2)(Q(n) - EQ(n)) does not converge to a Gaussian distribution, We provide conditions on the c(j)'s and on <(e ta)over cap> for the quadratic form Q(n), adequately normalized and ra ndomly centered, to converge to a stable law of index alpha, 1 < alpha < 2, as n tends to infinity.