Linear processes are nearly Gaussian

Abstract

LetUdenote the set of all integers, and suppose thatY= {Yu;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= {au;u∈U} be a sequence of real numbers with Σuau2= 1. ThenXu= ΣwawYu–wdefines a stationary linear processX= {Xu; u ɛ U} withE(Xu) = 0,E(Xu2) = 1 foru∊U. LetF(·) be the d.f. ofX0.We prove that if maxu|au| is small, then (i) for eachw, Xwis close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy≦gmaxu|au| where Φ(·) is the standard Gaussian d.f., andgdepends only onG(·); (ii) for each finite set (w1, …wn), (Xw1, …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.