ARCH-type bilinear models with double long memory

We discuss the covariance structure and long-memory properties of stationary solutions of the bilinear equation X-t = zeta(t)A(t) + B-t (star), where zeta(t), t epsilon Z are standard i.i.d. r.v.'s, and A(t),B-t, are moving averages in X-s, s < t. Stationary solution of (star) is obtained as an orthogonal Volterra expansion. In the case A(t) = 1, X-t is the classical AR(infinity) process, while B-t equivalent to 0 gives the LARCH model studied by Giraitis et al. (Ann. Appl. Probab. 10 (2000) 1002). In the general case, X-t may exhibit long memory both in conditional mean and in conditional variance, with arbitrary fractional parameters 0 < d(1) < 1 and 0 < d(2) < 1/2 respectively. We also discuss the hyperbolic decay of auto- and/or cross-covariances of X and and the asymptotic distribution of the corresponding partial sums' processes. (C) 2002 Elsevier Science B.V. All rights reserved.