Journal of the
Korean Mathematical Society JKMS

For a double array of random elements $\{X_{mn};m\ge1,n\ge 1\}$ in a $p$-uniformly smooth Banach space, $\{b_{mn};m\geq 1, n\geq 1\}$ is an array of positive numbers, convergence of double random series $\sum_{m=1}^{\infty}\!\sum_{n=1}^{\infty}\!X_{mn}$, $\sum_{m=1}^{\infty}\!\sum_{n=1}^{\infty}b_{mn}^{-1}X_{mn}$ and strong law of large numbers $$b_{mn}^{-1}\sum_{i=1}^m\sum_{j=1}^nX_{ij}\to 0 \mbox{ as } m\wedge n\to\infty$$ are established.

Keywords: convergence of double random series, strong laws of large numbers, $p$-uniformly smooth Banach spaces, double array of random elements

MSC numbers: 60F15, 60B12