El. knyga: Random Processes by Example [World Scientific e-book]

Lifshits helps advanced graduate students and researchers in pure or applied mathematics become familiar with a wide class of key random processes; understand how probability theory works in an important applied problem; and recognize the variety of limit theorems, especially for random processes. Among the tools he describes are Gaussian processes; independently scattered measures; stochastic integrals; and compound Poisson, infinitely divisible, and stable distributions. The book could be used in a one-semester advanced course. Annotation ©2014 Ringgold, Inc., Portland, OR (protoview.com)

This volume first introduces the mathematical tools necessary for understanding and working with a broad class of applied stochastic models. The toolbox includes Gaussian processes, independently scattered measures such as Gaussian white noise and Poisson random measures, stochastic integrals, compound Poisson, infinitely divisible and stable distributions and processes.Next, it illustrates general concepts by handling a transparent but rich example of a “teletraffic model”. A minor tuning of a few parameters of the model leads to different workload regimes, including Wiener process, fractional Brownian motion and stable Levy process. The simplicity of the dependence mechanism used in the model enables us to get a clear understanding of long and short range dependence phenomena. The model also shows how light or heavy distribution tails lead to continuous processes or to processes with jumps in the limiting regime. Finally, in this volume, readers will find discussions on the multivariate extensions that admit completely different applied interpretations.The reader will quickly become familiar with key concepts that form a language for many major probabilistic models of real world phenomena but are often neglected in more traditional courses of stochastic processes.