Nonparametric estimation in mixing sequences of random variables

Abstract

Let X1,X2,… be random variables defined on (Ω,A,P) and taking values in Rt, t≥1. Suppose that the sequence {Xj},j≥1, is strictly stationary and φi-mixing, for some i=1,…,4, and let ƒ be the probability density function of X1. Let ƒ̂n(x) be the usual kernel estimate of ƒ(x), xϵRt. Under certain conditions, it is shown that f̂n(x) is a strongly consistent estimate of ƒ(x). Under some additional conditions, this consistency is shown to be uniform over certain sets extending over all of Rt as the sample size n tends to infinity. These results, specialized to certain Markov processes, provide strongly cosistent estimates, as well as uniformly, as above, strongly consistent estimates, for the initial, the 2t-variate joint and the transition probability density functions. Finally, a uniformly, in the above sense, strongly consistent estimate is obtained for the one-step transition distribution function of the process.