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, = ζtAt + Bt,(*), where ζt, t ∈ Z are standard i.i.d. r.v.'s, and At,Bt are moving averages in Xs, s < t. Stationary solution of (*) is obtained as an orthogonal Volterra expansion. In the case At ≡ 1, Xt is the classical AR(oo) process, while Bt ≡ 0 gives the LARCH model studied by Giraitis et al. (Ann. Appl. Probab. 10 (2000) 1002). In the general case, X, may exhibit long memory both in conditional mean and in conditional variance, with arbitrary fractional parameters 0 < d1 < 1/2 and 0 < d2 < 1/2, respectively. We also discuss the hyperbolic decay of auto- and/or cross-covariances of Xt and X2t and the asymptotic distribution of the corresponding partial sums' processes.