On the uniform complete convergence of density function estimates

Letf be a uniformly continuous density function. LetW be a non-negative weight function which is continuous on its compact support [a, b] and ∫ _a ^b W(x)dx=1. The complete convergence of $$mathop {sup }limits_{ - infty< s< infty } left| {frac{1}{{nbleft( n right)}}sumlimits_{k - 1}^n {Wleft( {frac{{s - X_k }}{{bleft( n right)}}} right)} - fleft( s right)} right|$$ to zero is obtained under varying conditions on the bandwidthsb(n), support or moments off, and smoothness ofW. For example, one choice of the weight function for these results is Epanechnikov's optimal function andnb ²(n)>n ^δ for some δ>0. The uniform complete convergence of the mode estimate is also considered.