On Strong Invariance Principles Under Dependence Assumptions

Abstract

Strong invariance principles with other of approximation $O(t^{1/2-\kappa})$ are obtained for sequences of dependent random variables. The basic dependence assumptions include various generalizations of martingales such as asymptotic martingales (amarts), semiamarts, and mixingales as well as processes characterized by a condition on the Doleans measure. Provided the partial sum process is uniformly integrable, also martingales in the limit and games fairer with time are included. Sufficient conditions for linear growth of the covariance function of the partial sums are given.