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 d - 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