It is known that if $(S\_i= \sum\_{j \leqslant i }X\_i, \mathcal F\_i)$ is a martingale, 
 then for each  

 $
\beta>1$, $\delta\in (0,\beta-1)$ and $\lambda>0$, and each integer $N \geqslant 1$, the inequality 
$$\tag{\*} 
 \mu\left\{\max\_{1\leqslant i\leqslant N}|S\_i|>\beta\lambda\right\}
 \leqslant \frac{\delta^2}{(\beta-\delta-1)^2}\mu\left\{\max\_{1\leqslant i\leqslant N}|S\_i|>\lambda \right\}+
 \\
 +\mu\left\{ \sum\_{i=1}^N \mathbb E [X\_i^2 \mid \mathcal F\_{i-1}]>\delta^2\lambda^2\right\}
 +\mu \left\{\max\_{1\leqslant i\leqslant N}|X\_i| > \delta\lambda^2\right\} 
$$
takes place (see for instance Hall and Heyde *Martingale limit theory and its applications*, 
page 28). One can deduce the upper bound for $\mathbb E\max\_{1\leqslant i\leqslant N}|S\_i|^p$
in Burkholder's inequality from (\*). 


Now, we are looking for multidimensional extensions. Below I present the so-called orthomartingales, but I will settle for any form of martingale difference random field. 


Now, assume that $( \mathcal F\_{n\_1,n\_2 })\_{n\_1,n\_2 \in\mathbb Z }$ is a commuting filtration, 
that is, 
\begin{equation}
 \mathbb E\left[ \mathbb E[X\mid \mathcal F\_{n\_1,n\_2} ]\mid \mathcal F\_{n'\_1,n'\_2}\right]
 =\mathbb E\left[X\mid \mathcal F\_{\min\{ n\_1,n'\_1\},\min\{ n\_2,n'\_2\}} \right]
\end{equation}
for each integrable random variable $X$ and each $(n\_1,n\_2), (n'\_1,n'\_2)$. We denote 
$\mathcal F\_{i,\infty}$ (respectively 
$\mathcal F\_{\infty,j}$) the $ \sigma$-algebra generated by $\bigcup\_{j\in \mathbb Z}\mathcal F\_{i,j}$ 
(resp. $\bigcup\_{i\in \mathbb Z}\mathcal F\_{i,j}$ ). 


Assume that the random field $(X\_{i,j})\_{(i,j)\in\mathbb Z^2}$ is adapted to the filtration 
$( \mathcal F\_{n\_1,n\_2 })\_{n\_1,n\_2 \in\mathbb Z }$ an that it satisfies 
$ \mathbb E[X\_{i,j} \mid \mathcal F\_{i-1,\infty}]=X\_{i-1,j}$ 
and $ \mathbb E[X\_{i,j} \mid \mathcal F\_{\infty,j-1}]=X\_{i,j-1}$. The process $(\sum\_{i\_1=1}^{n\_2}\sum\_{i\_2=1}^{n\_2}X\_{i\_1,i\_2} )\_{n\_1,n\_2\in\mathbb N^2 }$
is called an *orthomartingale*, see Khosnevisian's book *Multiparameter processes*. 




---


**Question:** Is there (or is it possible to establish) an 
inequality in the spirit of (\*) for 
orthomartingale random fields?




---


Such an inequality would allow to derive a moment bound for 
$\max\_{i\leqslant n\_1,j\leqslant n\_2}|S\_{i,j}| ^p$. This can be derived by an application 
of Doob's inequality and Burkholder's one dimensional inequality.


I am not sure that (\*) holds for submartingales; therefore we cannot apply it directly 
to $\max\_{1\leqslant j\leqslant n\_2}| S\_{i,j}|$ for a fixed $i\leqslant n\_1$. 


The main difficulty is that we do not have a useful total order on $\mathbb N^2$. Also, 
it seems difficult to identify what would play the role of the quadratic variances for $d=1$. 



Tail inequality for orthomartingales/martingale difference random fields
Actually, an extension of (\*) is a way to establish Burkholder's inequality 
or [Nagaev's inequality](http://link.springer.com/article/10.1023%2FA%3A1025814306357). For general ortho-martingale random fields, this is a difficult task. 


However, if we assume that the random field is strictly stationary and the filtration completely commuting, then we can establish probability inequalities (see [Subsection 3.2](http://arxiv.org/abs/1603.00432)) and recover moment inequalities. 


