ON THE ASYMPTOTIC-EXPANSION OF THE EMPIRICAL PROCESS OF LONG-MEMORY MOVING AVERAGES

Let X(n) = Sigma(i=1)(infinity)a(i) epsilon(n-i), where the epsilon(i) are iid with mean 0 and finite fourth moment and the a(i) are regular ly varying with index -beta where beta is an element of (1/2, 1) so th at (X(n)) has long-range dependence. This covers an important class of the fractional ARIMA process. For r greater than or equal to 0, let Y -N,Y-r = Sigma(n=1)(N) Sigma(1 less than or equal to j1<...<jr)II(s=1) (r)a(js)epsilon(n-js), Y-N,Y-0 = N, sigma(N,r)(2) = Var(Y-N,Y-r) and F -(r) = rth derivative of the distribution function of X(n). The Y-N,Y- r are uncorrelated and are stochastically decreasing in r. For any pos itive integer p < (2 beta - 1)(-1), it is shown under mild regularity conditions that, with probability 1, [GRAPHICS] uniformly for all x is an element of R For All 0 < lambda < (beta - 1/2) boolean AND (1/2 - p(beta - 1/2)). This generalizes a host of existing results and provid es the vehicle for a number of statistical applications.