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 Sigma(m)(i=1) Sigma(n)(j=1) 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 Moricz et al. [J. Theoret. Probab., 21, 660-671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [ Ann. Probab., 6, 469-482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.