Abstract. Fazekas and Klesov (2000) found conditions for almost sure convergence rates in the law of large numbers that effectively can be applied if maximal inequalities are available. In the spirit of Móricz (1976), we aim at using those conditions in a weakly dependent framework, and this trick is proved to be quite efficient, first in the standard law of large numbers and second in the nonparametric estimation context where rates of convergence of the density kernel estimates are also obtained.

AMS Subject Classification (1991): 60F15, 60F99, 60G10, 62G07

Keyword(s): dependence, mixing strong law of large numbers, rate of convergence, kernel estimators

Received December 11, 2008, and in revised form May 10, 2010. (Registered under 79/2008.)