Functional Limit Theorems for Dependent Variables

Conditions are given for a sequence of stochastic processes derived from row sums of an array of dependent random variables to converge to a process with stationary, independent increments or to a process with continuous paths. We also discuss when row maxima converge to an extremal process. The first result is a generalization of the classical results for independent random variables. The second result gives general conditions for convergence to processes which can be obtained from Brownian motion by a random change of time. This result is used to give a unified development of most of the martingale central limit theorems in the literature. An important aspect of our methods is that after the initial result is shown, we can avoid any further consideration of tightness.