Abstract
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 Σ m i=1 Σ 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 Móricz 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.
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Stadtmüller, U., Van Thanh, L. On the strong limit theorems for double arrays of blockwise M-dependent random variables. Acta. Math. Sin.-English Ser. 27, 1923–1934 (2011). https://doi.org/10.1007/s10114-011-0110-z
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DOI: https://doi.org/10.1007/s10114-011-0110-z
Keywords
- Blockwise M-dependent random variables
- strong law of large numbers
- double arrays of random variables
- almost sure convergence