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, fraction.

Preface; Acknowledgments; Contents; 1. Preliminaries; 1 Random Variables: a Summary; 1.1 Probability space, events, independence; 1.2 Random variables and their distributions; 1.3 Expectation; 1.4 Inequalities based on expectation; 1.5 Variance; 1.6 Covariance, correlation coefficient; 1.7 Complex-valued random variables; 1.8 Characteristic functions; 1.9 Convergence of random variables; 2 From Poisson to Stable Variables; 2.1 Compound Poisson variables; 2.2 Limits of compound Poisson variables; 2.3 Amystery at zero; 2.4 Infinitely divisible random variables; 2.5 Stable variables

3 Limit Theorems for Sums and Domains of Attraction4 Random Vectors; 4.1 Definition; 4.2 Convergence of random vectors; 4.3 Gaussian vectors; 4.4 Multivariate CLT; 4.5 Stable vectors; 2. Random Processes; 5 Random Processes: Main Classes; 6 Examples of Gaussian Random Processes; 6.1 Wiener process; 6.2 Brownian bridge; 6.3 Ornstein-Uhlenbeck process; 6.4 Fractional Brownian motion; 6.5 Brownian sheet; 6.6 Levy's Brownian function; 6.7 Further extensions; 7 Random Measures and Stochastic Integrals; 7.1 Random measures with uncorrelated values; 7.2 Gaussian white noise

7.3 Integral representations7.3.1 Wi…