Baum-Katz type theorems for martingale arrays
Résumé
We show convergence rates in the law of large numbers for martingale arrays. The results extend the classical theorems of Baum and Katz (1965) for sums of independent and identically distributed (i.i.d.) random variables. They improve a result of Ghosal and Chandra (1998) for martingale arrays, and generalize a result of Alsmeyer (1990) for a single martingale. As an application, we obtain a new theorem about the convergence rate of Cesàro summation of identically distributed random variables.