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 ContentsFair 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 ZRecurrence and Transience of General Random WalksRandom Walks and AnalysisDonsker's Invariance Principle and Brownian Motion
