Random Processes by Example

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

Contents:

• Preliminaries:

  • Random Variables: A Summary
  • From Poisson to Stable Variables
  • Limit Theorems for Sums and Domains of Attraction
  • Random Vectors

• Random Processes:

  • Random Processes: Main Classes
  • Examples of Gaussian Random Processes
  • Random Measures and Stochastic Integrals
  • Limit Theorems for Poisson Integrals
  • Lévy Processes
  • Spectral Representations
  • Convergence of Random Processes

• Teletraffic Models:

  • A Model of Service System
  • Limit Theorems for the Workload
  • Micropulse Model
  • Spacial Extensions

Readership: Graduate students and researchers in probability & statistics.
Key Features:

• A thorough choice of self-contained material packed in a small volume enabling the reader to focus on really important issues and reach the frontline of research in a pretty short time
• Main examples explaining the theory originating from the modern research field
• Handling full scale examples in an in-depth manner (unusual for a textbook) brings in a touch of research work in an otherwise routine learning method