This is not available 009535

This thesis makes some contributions to the study of stationary processes, with a view towards applications to time series analysis. Ever since the work of Gordin [15], martingale approximations have been a useful tool to study stationary random walks (random walks with stationary increments). One highlight of this dissertation is a useful necessary and sufficient condition for martingale approximations; this condition provides new insight unifying several recent developments on this topic in the literature. As an application of martingale approximations, we derive a fairly sharp sufficient condition for the law of the iterated logarithm, including the functional form, and an improvement of the conditional central limit theorem of Maxwell and Woodroofe [30]. For statistical applications, we consider the problem of estimating a monotone trend nonparametrically for time series data. The asymptotic distributions of isotonic estimators are analyzed, and the accuracy of the approximations are studied numerically. Estimation of the end point value is a main focus because of the practical importance and mathematical difficulty.