Limit theorems for bivariate Appell polynomials. Part II: Non-central limit theorems

Let (X-t, t is an element of Z) be a linear sequence with non-Gaussian innovations and a spectral density which varies regularly at low frequencies. This includes situations, known as strong (or long-range) dependence, where the spectral density diverges at the origin. We study quadratic forms of bivariate Appell polynomials of the sequence (X-t) and provide general conditions for these quadratic forms, adequately normalized, to converge to a non-Gaussian distribution, We consider, in particular, circumstances where strong and weak dependence interact. The limit is expressed in terms of multiple Wiener-Ito integrals involving correlated Gaussian measures.