Linear processes are nearly Gaussian

Abstract

Let U denote the set of all integers, and suppose that Y = {Y_u; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {a_u; u ∈ U} be a sequence of real numbers with Σ_ua_u² = 1. Then X_u = Σ_wa_wY_u–w defines a stationary linear process X = {X_u; u ɛ U} with E(X_u) = 0, E(X_u²) = 1 for u ∊ U. Let F(·) be the d.f. of X₀. We prove that if max_u |a_u| is small, then (i) for each w, X_w is close to Gaussian in the sense that ∫^∞_−∞(F(y) − Φ(y))²dy ≦ g max_u |a_u | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w₁, … w_n), (X_w1, … X_wn) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of max_u|a_u| to the bandwidth of the filter a is studied.