Stationary Random Walks and Isotonic Regression: with Prediction for Time Series

The study of stationary processes is an important topic both in the probability and statistics literatures. This monograph continues the investigation along the classical lines, but with a modern viewpoint. For example, to study the behavior of the partial sums, it features the interplay between ergodic theory and probability from an operator-theoretical point of view. As a consequence, the results tend to be much more refined, and nearly optimal on many occasions. With the tools developed to understand the probabilistic behavior of the processes, one statistical application is considered in the context of global warming. The goal is to nonparametrically estimate the trend of a time series under monotonicity assumptions. Some interesting features in time series analysis are carefully explored.