Strong Invariance Principles for Dependent Random Variables

We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions. 1. Introduction. Strong laws of large numbers (SLLN), laws of the iterated logarithm (LIL), central limit theorems (CLT), strong invariance principles (SIP) and other variants of limit theorems have been extensively studied. Many deep results have been obtained for independent and identically distributed (i.i.d.) random variables. With various weak dependence conditions , some of the results obtained under the i.i.d. assumption have been generalized to dependent random variables. The primary goal of the paper is to establish a SIP for stationary processes. To this end, we shall develop some moment and maximal inequalities. As our basic tool, a new version of martingale approximation is provided. The martingale method was first applied in Gordin [21] and Gordin and Lifsic [22] and it has undergone substantial improvements. For recent contributions see Merlevède and Peligrad [38], Wu and Woodroofe [67] and Peligrad and Utev [40], where the central limit theory and weak convergence problems are considered. The approximation scheme acts as a bridge which connects stationary processes and martingales. One can then apply results from martingale theory (Chow and Teicher [7] and Hall and Heyde [23]),