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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 $d=1$.

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Actually, an extension of (*) is a way to establish Burkholder's inequality or Nagaev's inequality. 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) and recover moment inequalities.

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