Strong consistency of nearest neighbor kernel regression estimation for stationary dependent samples

Under quite mild conditions on k n , the strong consistency is proved for the nearest neighbor density, the nearest neighbor kernel regression and the modified nearest neighbor kernel regression of an α mixing stationary sequence in time series context. The condition imposed on the mixing coefficients is ∑∞j=1j a-1 α(j) 1-1/ν <∞(a>1, ν>1)  or ∑∞j=1j a-1 α(j)<∞(a>1), which is simple and weak.