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.