Note on the uniform convergence of density estimates for mixing random variables

On the basis of the random variables X1,...,Xn drawn from the (strictly) stationary and [phi]i-mixing (for some i = 1,...,4) stochastic process {Xn}, n [greater-or-equal, slanted] 1, a uniformly strongly consistent estimate of the (common) probability density function of the X's is constructed. For the case that the underlying process is also Markovian, uniformly strongly consistent estimates are constructed for the initial, the (X1, X2)-joint and the transition probability density functions of the process.