DISTANT LONG-RANGE DEPENDENT SUMS AND REGRESSION ESTIMATION

Consider a stationary sequence G(Z(0)), G(Z(1)), ... , where G(.) is a Borel function and Z(0), Z(1), ... is a sequence of standard normal variables with covariance function E(Z(0)Z(j)) = j(-alpha)L(j), j = 1, 2, ... ,where E(G(Z(0))) = 0, E(G(2)(Z(0))) < infinity, 0 < alpha < 1 and L(.) varies slowly at infinity. Let S-n(t) = Sigma(j=0)([m]-1)G(Z(j)), t greater than or equal to 0, be the associated partial-sum process. The main result is that for any fixed k is an element of N and 0 < b < infinity, a suitable norming sequence a(n) > 0 and sequences of gap-lengths l(1,n), ... , l(k,n) such that l(1,n) --> infinity and l(j,n) - l(j-1,n) --> infinity, j = 2, ... , k, arbitrary slowly, the vector process (S-n(t(0)), S-n(l(1,n) + t(1)) - S-n(l(1,n), ... , S-n(l(k,n) + t(k)) - S-n(l(k,n)/a(n), 0 less than or equal to t(0), t(1), ... , t(k) less than or equal to 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(Z(j)), j = 0, 1, ... .