On generalization of the Nagaev-Fuk inequalities for a class of random fields (Q1920862)

The paper deals with discrete multiparameter fields \(\{S_n,F_n\}\), \(n\in\mathbb{Z}^d_+\), \(d\geq2\), satisfying some kind of supermartingale property, that is stronger than usual supermartingale property and weaker than strong supermartingale property (this property coincides with the strong one when \(d=2\)). The main theorems present some estimates of the probability \(P\{\max_{k\leq n} S_k\geq x\}\). Incidentally, the author compares different forms of conditional independent property of multiparameter \(\sigma\)-fields.