The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Abstract

Consider the stationary sequenceX 1=G(Z 1),X 2=G(Z 2),..., whereG(·) is an arbitrary Borel function andZ 1,Z 2,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z 1 Z k+1) satisfyingr(0)=1 and ∑ =1 ∞ |r(k)| m < ∞, where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X n }, the integerm is the Hermite rank of the family {I{G(·)≦ x} −F(x):x∈R}. LetF n (·) be the empirical distribution function ofX 1,...,X n . We prove that, asn→∞, the empirical processn 1/2{F n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[−∞,∞].