On multivariate variable‐kernel density estimates for time series

Let X 1 be a strictly stationary multiple time series with values in R d and with a common density f . Let X 1 ,.,.,X n , be n consecutive observations of X 1 . Let k = k n , be a sequence of positive integers, and let H ni be the distance from X i to its k th nearest neighbour among X j , j i . The multivariate variable‐kernel estimate f n , of f is defined by\documentclass{article}\pagestyle{empty}\begin{document}$$ f_n ({\bf x}) = n^{ - 1} \sum\limits_{i = 1}^n {H_{ni}^{ - d} K} \left( {\frac{{{\bf X}_i - {\bf x}}}{{H_{ni} }}} \right), $$\end{document}where K is a given density. The complete convergence of f n , to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time‐series models.