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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!

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    $\begingroup$ I found the notes French for Mathematicians: A linguistic approach by Joël Bellaïche very helpful. He argues that someone who knows English can learn how to read a mathematical text in French without too much effort. $\endgroup$
    – gsa
    Jul 21, 2020 at 12:50
  • $\begingroup$ @gsa thanks for the tip $\endgroup$
    – user123124
    Jul 21, 2020 at 12:52

1 Answer 1

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$\newcommand\Om\Omega$ $\newcommand\F{\mathcal F}$ $\newcommand\M{\mathcal M}$ 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.

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  • $\begingroup$ The notes contained what I wanted, isnt theorem 1.8 more general? $\endgroup$
    – user123124
    Jul 21, 2020 at 17:34
  • $\begingroup$ @user1 : I don't understand your comment. What notes? What theorem 1.8? Where? $\endgroup$ Jul 21, 2020 at 18:00
  • $\begingroup$ gatsby.ucl.ac.uk/~porbanz/teaching/G6106S15/… page 6 $\endgroup$
    – user123124
    Jul 21, 2020 at 18:23
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    $\begingroup$ @user1 : Yes, Theorem 1.8 in the notes linked to in my answer is more general than Theorem 2 translated in my answer, in two ways: (i) the index set in Theorem 1.8 can be any directed set and (ii) the integrability condition in Theorem 2 is of course narrower than the general uniform integrability condition in Theorem 1.8 in the notes linked to in my answer. $\endgroup$ Jul 21, 2020 at 18:31

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