Strong consistency and rates for recursive nonparametric conditional probability density estimates under (α, β)-mixing conditions

Abstract

Let { X j : j ⩾ 1} be a real-valued stationary process. Recursive kernel estimators of the joint probability density functions, and of conditional probability density functions of X j , given past behavior, are considered. Their strong consistency, along with rates, are given for process { X j ; j ⩾ 1} satisfying (α, β)-mixing conditions. Here, we improve the rates of a.s. convergence in Masry (1987, 1989) without imposing considerably faster rate of decay on the mixing coefficients.