Asymptotic Behavior of Continuous Set-Indexed Martingales

In this paper, following the Knightʹs approach, we solve a convergence problem for set-indexed martingales. For this purpose, we first define a tightness criterion for set-indexed continuous processes. The core of this characterization is connected with a weaker definition of continuity and hence the use of the corresponding topology, and with the fact that indices take values in a semilattice of closed subsets. Then, we give an effective tightness criterion by means of an estimate for a majorizing measure defined on the space. We finally prove under this set-indexed framework a theorem similar to the Knightʹs.