Linear processes are nearly Gaussian

Abstract

LetUdenote the set of all integers, and suppose thatY= {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(·). Leta= {a_u;u∈U} be a sequence of real numbers with Σ_ua_u²= 1. ThenX_u= Σ_wa_wY_u–wdefines a stationary linear processX= {X_u; u ɛ U} withE(X_u) = 0,E(X_u²) = 1 foru∊U. LetF(·) be the d.f. ofX₀.We prove that if max_u|a_u| is small, then (i) for eachw, X_wis close to Gaussian in the sense that ∫^∞_−∞(F(y) − Φ(y))²dy≦gmax_u|a_u| where Φ(·) is the standard Gaussian d.f., andgdepends only onG(·); (ii) for each finite set (w₁, …w_n), (X_{w1, …Xwn ) is close to Gaussian in a similar sense; (iii) theprocess Xis close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filterais studied.}