LIMIT-THEOREMS FOR BIVARIATE APPELL POLYNOMIALS .1. CENTRAL LIMIT-THEOREMS

Consider the stationary linear process X(t) = Sigma(u=-infinity)(infin ity) a(t-u)xi(u), t is an element of Z, where {xi(u)} is an i.i.d. fin ite variance sequence. The spectral density of {X(t)} may diverge at t he origin (long-range dependence) or at any other frequency. Consider now the quadratic form Q(N) = Sigma(t,s=1)(N)b(t - s)P-m,P-n(X(t), X(s )), where P-m,P-n(X(t), X(s)) denotes a non-linear function (Appell po lynomial). We provide general conditions on the kernels b and a for N( -1/2)Q(N) to converge to a Gaussian distribution. We show that this co nvergence holds if b and a are not too badly behaved. However, the goo d behavior of one kernel may compensate for the bad behavior of the ot her. The conditions are formulated in the spectral domain.