Data-dependent kn-NN and kernel estimators consistent for arbitrary processes

Let X₁, X₂,... be an arbitrary random process taking values in a totally bounded subset of a separable metric space. Associated with X_i we observe Y_i drawn from an unknown conditional distribution F(y|X_i=x) with continuous regression function m(x)=E[Y|X=x]. The problem of interest is to estimate Y_n based on X_n and the data {(X_i, Y_i)}_i=1^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 X₁, X₂,.