Introduction to Random Processes; (translated by Birgit Rothinger)

Today, the theory of random processes represents a large field of mathematics with many different branches. This Introduction to the Theory of Random Processes applies mathematical models that are simple, but that have some importance for applications. The book starts with a treatment of homogeneous Markov processes with a countable number of states. The main topics are the ergodic theorem, the method of Kolmogorov's differential equations and Brownian motion, and the connecting link being the transition from Kolmogorov's differential-difference equations for random walk to a limit diffusion equation. The chapters that follow outline the foundations of stochastic analysis. They deal with random processes as curves in the space of random variables with the norm of quadratic mean. Random processes are then described by linear stochastic differential equations and their convergence behaviour is explored. The fundamentals of spectral analysis of stationary processes are considered and, finally, some special problems of estimation and filtration are discussed. In chapter 6 an attempt is made to apply direct probabilistic methods for sums of i.i.d. variables to a multi-server-system. As a complement, chapters 9 to 11 deal with nonlinear stochastic differential equations for diffusion processes.