Rates of convergence for the empirical distribution function and the empirical characteristic function of a broad class of linear processes

Abstract

This paper deals with uniform rates of convergence for the empirical distribution function and the empirical characteristic function for a broad class of stationary linear processes. In particular, the class X( n) = Σ i=0 ∞ δ( i) z( n−1) is considered under the conditions that (a) the disturbances z( n) are independent and identically distributed with a finite first absolute moment, (b) the distribution function F of X( n) has bounded density, and (c) the parameters δ( i) are bounded in absolute value by some function g which satisfies Σ i=1 ∞ ig( i) < ∞. It is proved that the empirical distribution function F ̂ (x) based on X( n), n = 1, …, N, converges to the theoretical distribution, uniformly in x, at a rate O(N −1 2 logN) a.s. For the empirical characteristic function a uniform (over a certain range of the argument) rate of convergence result is proved. A pointwise central limit theorem is also obtained. The above theorems constitute versions (weaker in part) of theorems that already exist in the i.i.d. case and for certain sequences of dependent random variables.