Introduction to Mathematical Tools

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.

Illustration with a Teletraffic Model

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 Lévy 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.

Dependence and Distribution Tails

The model also shows how light or heavy distribution tails lead to continuous Gaussian processes or to processes with jumps in the limiting regime.

Extensions and Key Concepts

Finally, in this volume, readers will find discussions on the multivariate extensions that admit a variety of 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.