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-[alpha]L(j), j = 1, 2, ..., where E(G(Z0)) = 0, E(G2(Z0)) < [infinity], 0 < [alpha] < 1 and L(·) varies slowly at infinity. Let Sn(t) = [summation operator][lower left corner]nt[right floor]-1j=0 G(Zj), t [greater-or-equal, slanted] 0, be the associated partial-sum process. The main result is that for any fixed and 0 < b < [infinity], a suitable norming sequence an > 0 and sequences of gap-lengths l1,n, ..., lk,n such that l1,n --> [infinity] and lj,n - lj-1,n --> [infinity], j = 2, ..., k, arbitrary slowly, the vector process , 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, ... .