On the strong limit theorems for double arrays of blockwise M-dependent random variables

For a double array of blockwise M-dependent random variables {X _mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? _i=1 ^m ?? _j=1 ⁿ X _ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X _mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X _mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.