On the rate of convergence in the strong law of large numbers for martingales

The aim of this note is to establish the Baum-Katz type rate of convergence in the Marcinkiewicz-Zygmund strong law of large numbers for martingales, which improves the recent works of Stoica [Series of moderate deviation probabilities for martingales, J. Math. Anal. Appl. 336 (2005), pp. 759-763; Baum-Katz-Nagaev type results for martingales, J. Math. Anal. Appl. 336 (2007), pp. 1489-1492; A note on the rate of convergence in the strong law of large numbers for martingales, J. Math. Anal. Appl. 381 (2011), pp. 910-913]. Furthermore, we also study some relevant limit behaviours for the uniform mixing process. Under some uniform mixing conditions, the sufficient and necessary condition of the convergence of the martingale series is established.