Linear processes are nearly Gaussian

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 u a u 2 = 1. Then X u = Σ w a w Y u – w defines a stationary linear process X = X u ; u ɛ U with E (X u) = 0, E (X u 2) = 1 for u ∊ U. Let F (·) be the d. f. of X 0. 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) ) 2 dy ≦ g max u | a u | where Φ (·) is the standard Gaussian d. f. , and g depends only on G (·) ; (ii) for each finite set (w 1, … w n), (X w 1, … X w n) 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.