For part of my work I am using k-NN regression for $\{X,Y\}$ be jointly stationary and ergodic Markov. I am looking for some results on the consistency of NN regression for estimating $E[Y|X=x]$, i.e., find $k$ nearest neighbors of $x$ among $X_1$,...,$X_n$ and make an empirical average over the corresponding $Y$s. In my literature review, I have found works by Yakowitz et.al. [1] that has shown the results when $Y_n$ conditioning on $X_n$ is independent of the previous values of $X$ and $Y$ (Like a hidden Markov model with $X$ to be hidden states). I have also checked Yakowitz et. al. [2] where the authors are investigating kernel and partitioning methods only.
I would appreciate it if someone can point me in the right direction or a result on the NN regression for estimating $E[Y|X=x]$ and non-iid data without assuming the hidden Markov structure explained above.
[1] Karlsson M, Yakowitz S. Nearest‐neighbor methods for nonparametric rainfall‐runoff forecasting. Water Resources Research. 1987 Jul;23(7):1300-8.
[2] Yakowitz S, Györfi L, Kieffer J, Morvai G. Strongly consistent nonparametric forecasting and regression for stationary ergodic sequences. Journal of multivariate analysis. 1999 Jan 1;71(1):24-41.