Multiparameter Subadditive Processes

Abstract

Let $N$ be the positive integers. We define a class of processes indexed by $N^r \times N^r$ which we call subadditive (when $r = 1$ our definition coincides with the usual one). Under a first moment condition we prove mean convergence of $x_{0t}/|\mathbf{t}|$ as each coordinate of $\mathbf{t} \rightarrow \infty$, where $|\mathbf{t}| = t_1 t_2 \cdots t_r$. If the process is strongly subadditive (a more restrictive condition) then the same first moment condition gives a.s. sectorial convergence. We conjecture (and verify in several cases) that an $L(\log L)^{r-1}$ integrability condition is sufficient to give unrestricted a.s. convergence.