Bulletin of the
Korean Mathematical Society BKMS

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