Does anyone know of a English translation of "Une inégalité pour martingales à indices multiples et ses applications" by Renzo Cairoli. Or could translate the statement of the martingale convergence theoerm and his definition of multiindex martingale.
Link; http://archive.numdam.org/article/SPS_1970__4__1_0.pdf
References where theorem is stated clearly also works!
$\newcommand\Om\Omega$ $\newcommand\F{\mathcal F}$ $\newcommand\M{\mathcal M}$ 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 $\Om=\prod\limits_j\Om_j$, $\F=\bigotimes\limits_j\F_j$, $P=\bigotimes\limits_j P_j$. 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 $r_1\le r'_1,\dots,r_m\le r'_m$.
We will denote by $\M$ the class of martingales $$(X_{r_1,\dots,r_m}, \F_{r_1}\otimes\cdots\otimes\F_{r_m})$$ 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 $(X_{n_1,\dots,n_m})\in\M$ 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 $\sup\limits_{n_1,\dots,n_m}E|X_{n_1,\dots,n_m}|^p<\infty$ for some $p> 1$), then the limit $$\lim_{n_1\to\infty,\dots,n_m\to\infty}X_{n_1,\dots,n_m}$$ exists (and is finite) a.s.
