English translation of "Une inégalité pour martingales à indices multiples et ses applications"

$newcommandOmOmega$ $newcommandF{mathcal F}$ $newcommandM{mathcal M}$ With the help from Google Translate:

Throughout the work $m$ is a fixed integer $ge2$ and $j$ runs through the integers from $1$ to $m$. For each $j$, $(Om_j,F_j,P_j)$ is a probability space. Set $Om=prodlimits_jOm_j$, $F=bigotimeslimits_jF_j$, $P=bigotimeslimits_j P_j$. The expectation with respect to $P$ will be denoted by $E$.

The processes that we are going to consider are (unless otherwise stated) real, defined on $(Om,F,P)$ and admitting as a set of indices a set of points with $m$ coordinates of which each coordinate traverses a countable subset of $mathbb R$. This set will be endowed with the relation $(r_1,dots,r_m)le(r'_1,dots,r'_m)$ if $r_1le r'_1,dots,r_mle r'_m$.

We will denote by $M$ the class of martingales $$(X_{r_1,dots,r_m}, F_{r_1}otimescdotsotimesF_{r_m})$$ relative to an increasing family of product [$sigma$-]fields contained in $F$.

Apparently, here the general definition of a martingale over a directed partially ordered index set is assumed; see e.g. Section Filtrations and martingales.

Theorem 2. If for the martingale $(X_{n_1,dots,n_m})inM$ we have $$sup_{n_1,dots,n_m}E{|X_{n_1,dots,n_m}|(log^+|X_{n_1,dots,n_m}|)^{m-1}}<infty$$ (therefore, in particular, if $suplimits_{n_1,dots,n_m}E|X_{n_1,dots,n_m}|^p<infty$ for some $p> 1$), then the limit $$lim_{n_1toinfty,dots,n_mtoinfty}X_{n_1,dots,n_m}$$ exists (and is finite) a.s.