Mean convergence theorems and weak laws of large numbers for double arrays of random elements in Banach spaces
Bull. Korean Math. Soc. 2010 Vol. 47, No. 3, 467-482
Published online May 1, 2010
Le Van Dung and Nguyen Duy Tien
Danang University of Education and National University of Hanoi
Abstract : For a double array of random elements $\{V_{mn};m\ge1,n\ge 1\}$ 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 $k_{mn}^{-\frac{1}{r}}\sum_{i=1}^{u_m}\sum_{j=1}^{v_n}(V_{ij}-E(V_{ij}|\mathcal{F}_{ij}))\to 0$ in $L_r\, (0 < r <2)$. The weak law results provide conditions for $k_{mn}^{-\frac{1}{r}}\sum_{i=1}^{T_m}\sum_{j=1}^{\tau_n}(V_{ij}-E(V_{ij}|\mathcal{F}_{ij}))\to 0$ in probability where $\{T_m; m\geq 1\}$ and $\{\tau_n; n\geq 1\}$ are sequences of positive integer-valued random variables, $\{k_{mn}; m\geq 1, n\geq 1\}$ is an array of positive integers. The sharpness of the results is illustrated by examples.
Keywords : martingale type $p$ Banach spaces, double arrays of random elements, weighted double sums, weak laws of large numbers, mean convergence theorem
MSC numbers : 60B11, 60B12, 60F15, 60F25, 60G42
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