UNIFORM STRONG CONVERGENCE AND RATES FOR THE KERNEL ESTIMATORS OF A DISTRIBUTION FUNCTION AN D A REGRESSION FUNCTION UNDER WEAKLY DEPENDENT ASSUMPTIONS

Let X₁, X₂, …be a sequence of random variables with unknown distribution function F（x）. A kernel estimator of F（x） was suggested by Yamato. Ohai （1988） considered the strong consistency and rates for the estimator under φ-mixing condition. In the paper, we study the uniform strong convergency of the estimator under φ-mixing and a-mixing assumptions and the rate of the uniform strong convergence for the estimator under α-mixing assumption. Our conditions are weaker than those of Ohai （1988） and some results are as same as i. i. d, case.
Again, lot （X₁, Y₁）, （X₂, Y₂）, … be a sequence of p-mixing random variables. We discuss the strong consistency and rates for recursivo kernel estimator of rogression function.