Distant long-range dependent sums and regression estimation

Consider a stationary sequence G(Z0), G(Z1), …, where G(·) is a Borel function and Z0, Z1, … is a sequence of standard normal variables with covariance function E(Z0Zj) = j−αL(j), j = 1, 2, …, where E(G(Z0)) = 0, E(G2(Z0)) < ∞, 0 < α < 1 and L(·) varies slowly at infinity. Let Sn(t) = ∑⌞nt⌋−1j=0 G(Zj), t ⩾ 0, be the associated partial-sum process. The main result is that for any fixed k ϵ N and 0 < b < ∞, a suitable norming sequence an > 0 and sequences of gap-lengths l1,n, …, lk,n such that l1,n → ∞ and lj,n − lj−1,n → ∞, j = 2, …, k, arbitrary slowly, the vector process (Sn(t0), Sn(l1,n + t1) − Sn(l1,n), …, Sn(lk,n + tk) − Sn(lk,n))an, 0 〈 t0, t1, …, tk 〈 b, converges in distribution in D[0, b]k+1 to the vector of k + 1 independent Hermite processes with a rank given by G(·). As an application, the asymptotic behavior of the finite-dimensional distributions of kernel estimators is determined in the fixed-design regression model with errors of the form G(Zj), j = 0, 1, … .