Data-dependent kn-NN 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 an appropriate data-dependent nearest neighbor estimator and show, with a very elementary proof, that it is consistent for every process X₁ ,X₂