Random processes by example / Mikhail Lifshits.

Format:

Book

Author/Creator:

Lifshits, Mikhail (Mikhail Anatoévich), 1956- author.

Language:

English

Subjects (All):

Stochastic processes--Mathematical models.

Stochastic processes.

Physical Description:

1 online resource (232 p.)

Place of Publication:

Singapore : World Scientific, 2014.

Language Note:

English

Summary:

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

Contents:

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 Wiener process; 7.3.2 Brownian bridge; 7.3.3 Fractional Brownian motion; 7.3.4 Riemann-Liouville processes and operators; 7.3.5 Brownian sheet; 7.3.6 Levy's Brownian function; 7.4 Poisson random measures and integrals; 7.5 Independently scattered stable random measures and integrals; 8 Limit Theorems for Poisson Integrals; 8.1 Convergence to the normal distribution; 8.2 Convergence to a stable distribution; 9 Levy Processes; 9.1 General Levy processes; 9.2 Compound Poisson processes; 9.3 Stable Levy processes; 10 Spectral Representations

10.1 Wide sense stationary processes10.2 Spectral representations; 10.3 Further extensions; 11 Convergence of Random Processes; 11.1 Finite-dimensional convergence; 11.2 Weak convergence; 11.2.1 Metric spaces: reminder; 11.2.2 Weak convergence; 11.2.3 Basic examples of weak convergence; 3. Teletraffic Models; 12 A Model of Service System; 12.1 Main assumptions on the service time and resource consummation; 12.2 Analysis of workload variance; 12.2.1 Weak dependence; 12.2.2 Long range dependence; 13 Limit Theorems for the Workload; 13.1 Centered and scaled workload process

13.2 Weak dependence: convergence to Wiener process13.3 Long range dependence: convergence to fBm; 13.4 Convergence to a stable Levy process; 13.4.1 Dominating resource distribution; 13.4.2 Dominating service duration; 13.5 Convergence to Telecom processes; 13.5.1 Convergence to a stable Telecom process; 13.5.2 Convergence to a Poisson Telecom process; 13.6 Handling "messengers from the past"; 14 Micropulse Model; 15 Spacial Extensions; 15.1 Spacial model; 15.2 Spacial noise integrals; 15.2.1 White noise integral; 15.2.2 Fractional noise integral; 15.3 Limit theorems for spacial load

15.3.1 Weak dependence: convergence to white noise integral

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-4522-29-5