Marcinkiewicz–Zygmund laws for Banach space valued random variables with multidimensional parameters

For finite-dimensional array $X(m)=X(m_1,\dots,m_k)$ of independent identically distributed Banach space valued random variables we consider sums $S(n)=S(n_1,\dots,n_k)$ of $X(m)$ over $m_i\in\{1,\dots,n_i\}$ $(i-1,\dots,k)$. Under some conditions on individual random variable $X$ and on the geometry of Banach space the strong law of large numbers for $S(n)$ and estimates for large deviations as $\max n_i\to\infty$ are obtained.