A Maximal Inequality and Tightness for Multiparameter Stochastic Processes

Abstract

A maximal inequality is proven for a process with a multiparameter time index and is used to prove tightness for a sequence of such processes. Applications include functional limit theorems for planar processes, which are both 1- and 2- martingales, for generalized U-statistics and for row and column exchangeable arrays. When extending functional limit theorems for processes indexed by R+ = [0, ∞) to processes indexed by a multidimensional time parameter, it is experienced that frequently the most delicate aspect of the generalization is the proof of tightness of the sequence of processes. The proof of convergence of the finite dimensional distributions can often be obtained through one-dimensional methods, whereas the proof of tightness requires techniques which are genuinely multidimensional. This issue has been resolved if the processes are all strong martingales, because as was proven, convergence of the finite dimensional distributions in fact implies tightness when the limiting process is continuous. However, requiring strong martingale structure means that this result has limited applicability. In this paper one extends the above result to processes with asymptotic i-martingale structure provided it is known that the limit of the finite dimensional distributions is that of a particular class of continuous processes, which includes all Gaussian processes with independent increments and bounded, continuous variance functions.