Strong consistency and rates for recursive nonparametric conditional probability density estimates under ([alpha], [beta])-mixing conditions

Let {Xj: j [greater-or-equal, slanted] 1} be a real-valued stationary process. Recursive kernel estimators of the joint probability density functions, and of conditional probability density functions of Xj, given past behavior, are considered. Their strong consistency, along with rates, are given for process {Xj; j [greater-or-equal, slanted] 1} satisfying ([alpha], [beta])-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.