Two results are presented concerning the consistency of the $k$-nearest neighbor regression estimate. We show that all modes of convergence in $L_1$ (in probability, almost sure, complete) are equivalent if the regression variable is bounded. Under the additional conditional $k/\log n \rightarrow \infty$ we also obtain the strong universal consistency of the estimate.
"On the Strong Universal Consistency of Nearest Neighbor Regression Function Estimates." Ann. Statist. 22 (3) 1371 - 1385, September, 1994. https://doi.org/10.1214/aos/1176325633