Weak convergence and relative compactness of martingale processes with applications to some nonparametric statistics

For forward and reverse martingale processes, weak convergence to appropriate stochastic (but, not necessarily, Wiener) processes is studied. In particular, it is shown that martingale processes are tight under a uniformly integrability condition, and also, convergence of finite dimensional distributions satisfying certain mild conditions implies the compactness of such processes. The theory is illustrated with the aid of a class of U-statistics and von Mises' differentiable statistical functions which need not be stationary of order zero. Weak convergence of the classical Cramér-von Mises goodness-of-fit statistic is also considered. The case of martingales with random indices is studied at the end.