Exponential estimates for two-parameter martingales (Q1097575)

The paper is devoted to the exponential estimates for distributions of continuous two-parameter martingales. More precisely, the probability \(P\{\sup_{t\leq T}| x_ t| \geq \alpha T_ 1T_ 2\}\) is estimated by exponentials. The estimates are given for those martingales \(x_ t\) that have ``one-parameter'' square functions of bounded growth: \(<x,x>^ i_ t\leq Ct_ 1t_ 2\), \(i=1\) or \(i=2\), \(t=(t_ 1,t_ 2)\in R^ 2_+\), and also for martingales that have ``two-parameter'' square functions of bounded growth: \[ [x,x]_ t\leq C_ 1t_ 1t_ 2+C_ 2t^ 2_ 1t^ 2_ 2,\quad C_ i\geq 0,\quad i=1,2. \]