Data-dependent k/sub n/-NN and kernel estimators consistent for arbitrary processes

Let X/sub 1/, X/sub 2/,... be an arbitrary random process taking values in a totally bounded subset of a separable metric space. Associated with X/sub i/ we observe Y/sub i/ drawn from an unknown conditional distribution F(y|X/sub i/=x) with continuous regression function m(x)=E[Y|X=x]. The problem of interest is to estimate Y/sub n/ based on X/sub n/ and the data {(X/sub i/, Y/sub i/)}/sub i=1//sup n-1/. We construct appropriate data-dependent nearest neighbor and kernel estimators and show, with a very elementary proof, that these are consistent for every process X/sub 1/, X/sub 2/,.