Rate of convergence in the strong law of large numbers for martingales

A note on the rate of convergence in the strong law of large numbers for martingales

“…Łagodowski and Rychlik studied in [10] the convergence of the series (1) when {X n , n 1} is a martingale difference sequence (in short MDS); in particular, if p = r = 2, their results prove that the series (1) converges for any MDS bounded in L 2 , i.e., satisfying sup n 1 E( X 2 n ) < ∞. Stoica proved in [15] that, if 0 < r < 2 < p, then the series (1) converges for any MDS bounded in L p , i.e., satisfying sup n 1 E(| X n | p ) < ∞.…”

Convergence rates in the SLLN for some classes of dependent random fields

“…Most results concerning rate of convergence in the Marcinkiewicz SLLN for martingales were obtained in the onedimensional case. We refer to Lagodowski and Rychlik [11], Elton [5], Lesigne and Volny [13], Stoica [26] and Ghosal and Chandra [7] for martingale arrays.…”

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

“…Łagodowski and Rychlik [12] studied the convergence of the series in (1.3) when fX n ; n $ 1} is a martingale difference sequence, in particular, if p ¼ r ¼ 2, their results show that the series in (1.3) converges for any martingale difference sequence bounded in L 2 (i.e. satisfying sup n$1 EX 2 n , 1).…”