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!



English translation of "Une inégalité pour martingales à indices multiples et ses applications"
$\newcommand\Om\Omega$ $\newcommand\F{\mathcal F}$ $\newcommand\M{\mathcal M}$
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](http://www.gatsby.ucl.ac.uk/%7Eporbanz/teaching/G6106S15/Chapter1_27Jan15.pdf).



> 
> **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.
> 
> 
> 


