The M-Z-B Inequality for Multidimensionally Indexed Martingales and Its Application to Limit Theorems

In. this paper, under the hypothesis that a nonclecreasing array of multiclimensionally indexed -fields \left\\mathcalF_\vecn, \vecn \in N^r\right\ satisfies compatible condition instead of condition F₄ that introduced by Cairoli and Walsh, we prpve an extended Marcirikiewicz -Zygmund-Burkholder inequality for the multidimensionally indexed martingales. As an application we obtain a Brunk-Chung-Prohorov’s Strong Law of Large Numbers, and a result of complete convergence for multidimensionally indexed martingales.