Invariance principles for deconvolving kernel density estimation for stationary sequences of random variables

The deviation between the empirical process of the kernel-type deconvoluted estimated density and a Gaussian process of a stationary sequence in properly sup-norm metrics is studied. We propose a general method to obtain both weak and strong convergence results when the underlying sequence is strictly stationary alpha-mixing. The study of these approximations is conducted for both ordinary and super smooth errors. Further, by investigating the sup-norm distance of the empirical kernel-type deconvoluted process from its Gaussian counterpart, we find a rate of convergence of order n(-v)h(-(2+beta)), for some nu > 0 and beta > 1. (C) 2000 Elsevier Science B.V. All rights reserved.