Rene L. Schilling 
Martingale und Prozesse [PDF ebook] 

This is the third volume of the series 'Moderne Stochastik’ (Modern Stochastics). As a follow-up to the volume 'Wahrscheinlichkeit’ (Probability Theory) it gives an intrdouction to dynamical aspects of probability theory using stochastic processes in discrete time. The first part of the book covers discrete martingales – their convergenc behaviour, optional sampling and stopping, uniform integrability and essential martingale inequalities. The power of martingale techniques is illustrated in the chapters on applications of martingales in classical probability and on the Burkholder-Davis-Gundy inequalities. The second half of the book treats random walks on Zd and Rd, their fluctuation behaviour, recurrence and transience. The last two chapters give a brief introduction to probabilistic potential theory and an outlook of further developments: Brownian motion and Donsker”s invariance principle

Contents
Fair Play
Conditional Expectation
Martingale
Stopping and Localizing
Martingale Convergence
L2-Martingales
Uniformly Integrable Martingales
Some Classical Results of Probability
Elementary Inequalities for Martingales
The Burkholder–Davis–Gundy Inequalities
Random Walks on ℤd – the first steps
Fluctuations of Simple Random Walks on Z
Recurrence and Transience of General Random Walks
Random Walks and Analysis
Donsker”s Invariance Principle and Brownian Motion