MEAN CONVERGENCE THEOREMS AND WEAK LAWS OF LARGE NUMBERS FOR DOUBLE ARRAYS OF RANDOM ELEMENTS IN BANACH SPACES
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For a double array of random elements {Vmn;m ≥ 1, n ≥1} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which k−1r mn ∑um i=1∑vn j=1(Vij−E(Vij|Fij )) → 0 in Lr (0 < r < 2). The weak law results provide conditions for k −1rmn ∑Tm i=1∑τn j=1(Vij − E(Vij|Fij )) → 0 in probability where {Tm;m ≥ 1} and {τn; n ≥ 1} are sequences of positive integer-valued random variables, {kmn;m ≥ 1, n ≥ 1} is an array of positive integers. The sharpness of the results is illustrated by examples.
Abstract
For a double array of random elements {Vmn;m ≥ 1, n ≥1} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which k−1r mn ∑um i=1∑vn j=1(Vij−E(Vij|Fij )) → 0 in Lr (0 < r < 2). The weak law results provide conditions for k −1rmn ∑Tm i=1∑τn j=1(Vij − E(Vij|Fij )) → 0 in probability where {Tm;m ≥ 1} and {τn; n ≥ 1} are sequences of positive integer-valued random variables, {kmn;m ≥ 1, n ≥ 1} is an array of positive integers. The sharpness of the results is illustrated by examples.
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