The L/sub 1/ and L/sub 2/ strong consistency of recursive kernel density estimation from dependent samples

The L/sub 1/ and L/sub 2/ strong consistency of recursive kernel density estimators is established for mixing and ergodic stationary processes. Sharp rates of almost sure convergence are obtained in the L/sub 2/ case for mixing processes. In addition, the concept and properties of Hilbert space-valued mixingales are developed, and strong laws of large numbers are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>