MEAN CONVERGENCE THEOREMS AND WEAK LAWS OF LARGE NUMBERS FOR DOUBLE ARRAYS OF RANDOM ELEMENTS IN BANACH SPACES

Abstract

For a double array of random elements {<TEX>$V_{mn};m{\geq}1,\;n{\geq}1$</TEX>} 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 <TEX>$k_{mn}^{-\frac{1}{r}}\sum{{u_m}\atop{i=1}}\sum{{u_n}\atop{i=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$</TEX> in <TEX>$L_r$</TEX> (0 < r < 2). The weak law results provide conditions for <TEX>$k_{mn}^{-\frac{1}{r}}\sum{{T_m}\atop{i=1}}\sum{{\tau}_n\atop{j=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$</TEX> in probability where {<TEX>$T_m;m\;{\geq}1$</TEX>} and {<TEX>${\tau}_n;n\;{\geq}1$</TEX>} are sequences of positive integer-valued random variables, {<TEX>$k_{mn};m{{\geq}}1,\;n{\geq}1$</TEX>} is an array of positive integers. The sharpness of the results is illustrated by examples.