We derive both Azuma-Hoeffding and Burkholder-type inequalities for partial sums over a rectangular grid of dimension $d$ of a random field satisfying a weak dependency assumption of projective type: the difference between the expectation of an element of the random field and its conditional expectation given the rest of the field at a distance more than $\delta$ is bounded, in $L^p$ distance, by a known decreasing function of $\delta$. The analysis is based on the combination of a multi-scale approximation of random sums by martingale difference sequences, and of a careful decomposition of the domain. The obtained results extend previously known bounds under comparable hypotheses, and do not use the assumption of commuting filtrations.

翻译：对于定义在$d$维矩形网格上的随机场，我们推导了满足投影型弱相依假设的部分和的Azuma-Hoeffding型与Burkholder型不等式：该随机场中某个元素与其在距离超过$\delta$的其余场条件下的条件期望之间的$L^p$距离差，受限于一个已知的关于$\delta$的递减函数。该分析基于随机和多尺度近似为鞅差序列的方法，以及对区域的精细分解。所得结果在可比假设下扩展了先前已知的界限，且未使用滤子可交换性假设。