A strong law of large numbers for nonparametric regression

Abstract

Suppose (i1,n, ..., in,n) is permutation of (1, ..., n) for each positive integer n such that the order of the indices {1, h., n - 1} in the permutation corresponding to n - 1 is preserved. If {Zn} is a sequence of mean-zero random variables and {kn} is a sequence of positive integers with kn ? ? and kn n ? 0, we prove max1 ? j ? kn |?v = 1 j Ziv,n| kn ? 0 a.s. under a first moment-type assumption on {Zn} and appropriate conditions on the permutations and the growth rate of {kn}. The result is applied to prove strong consistency of nonparametric estimators of regression functions with heavy-tailed error distributions using the k-nearest neighbor and the unikform kernel methods under similar moment assumptions on the conditional distributions of the regressed variable.