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Models of Random Processes

A Handbook for Mathematicians and Engineers

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## Book Description

Devising and investigating random processes that describe mathematical models of phenomena is a major aspect of probability theory applications. Stochastic methods have penetrated into an unimaginably wide scope of problems encountered by researchers who need stochastic methods to solve problems and further their studies. This handbook supplies the knowledge you need on the modern theory of random processes.

Packed with methods, Models of Random Processes: A Handbook for Mathematicians and Engineers presents definitions and properties on such widespread processes as Poisson, Markov, semi-Markov, Gaussian, and branching processes, and on special processes such as cluster, self-exiting, double stochastic Poisson, Gauss-Poisson, and extremal processes occurring in a variety of different practical problems.

The handbook is based on an axiomatic definition of probability space, with strict definitions and constructions of random processes. Emphasis is placed on the constructive definition of each class of random processes, so that a process is explicitly defined by a sequence of independent random variables and can easily be implemented into the modelling.

Models of Random Processes: A Handbook for Mathematicians and Engineers will be useful to researchers, engineers, postgraduate students and teachers in the fields of mathematics, physics, engineering, operations research, system analysis, econometrics, and many others.

## Table of Contents

Definitions and General Properties of Random Processes

Definition of a Random Process via a Probability Measure over the Path Function Space

Definition of a Random Process via Multidimensional Distributions

Equivalence of Random Processes: Measurability, Separability

Stochastic Continuity

Definition of a Random Process via Mean and Covariance Functions

Mean-Square Continuity

Processes with Continuous Path Functions

Convergence of Random Processes

Invariance Principle

Ergodicity

References

Classification of Random Processes

State Space and a Parametric Set

Stationary in the Broad Sense Random Processes

Stationary Random Processes and Random Processes with Stationary Increments

Processes with Independent Increments

Point Processes: Memoryless Property

Markov Processes

Semi-Markov Processes

Renewal Processes (Recurrent Point Processes)

Regenerative Processes

Gaussian Processes

Martingales and Semi-Martingales

References

Discrete-Time Markov Chains

Definitions and Elementary Relations

Classification of States of a Markov Chain

Ergodic Theorems

Method of Generating Functions

Unbounded Random Walk

Bounded Random Walk

References

Main Classes of Constructively Defined Random Processes

Poisson Process

Continuous-Time Markov Chains

Markov Process in a Finite or Countable Phase-Space

Birth and Death Process

Application of Birth and Death Processes in Queueing Theory and Reliability Theory

Main Relations for a Semi-Markov Process

Applications of Semi-Markov Processes

Linewise Markov Process

Shot Noise Process

References

Random Processes with Independent Increments

Multidimensional Brownian Motion

Convergence of Sums of Infinitely Small Random Variables to the Process of Brownian Motion

Characterization of a General Process with Independent Increments

Properties of Path Functions

Convergence of Sums of Independent Random Variables to a Process with Independent Increments

Distribution of a Functional of the Process

References

Processes Associated with a Poisson Process

Some Properties of Point Processes: Probability Generating Functional

Cluster Processes

Secondary Processes

Self-Exiting and Mutually Exiting Point Processes

Doubly Stochastic Poisson Process (Cox Process)

Bivariate Poisson Process

Gauss-Poisson Process

References

Random Flows of Events

Main Definitions

Random Memoryless Flows

Stationary Flows

Flows with Bounded Aftereffect

Superposition of Point Processes

Limit Theorems for Thinned Flows

Marked Point Processes: Main Definitions

Palm Distribution

Processes with Embedded Marked Point Processes

Principle of Intensity Conservation

References

Classes of Constructively Defined Random Processes

Chains with Complete Connections

Processes Associated with Semi-Markov Process

Some Generalizations of Regenerative Processes

Cumulative Processes

Theory of Counters

Cascade Processes

Extremal Processes

Piecewise Linear Markov Processes

References

Some Special Classes of Processes

Stable Processes

Cauchy Processes

x2-Process, Bessel and Rayleigh Processes

Ornstein-Uhlenbeck Process

Subadditive Processes

Random Fields with Independent Increments

Periodic Random Processes

Random Processes Used by Description of Complex Systems

References

Stability of Random Processes

Stability of Complex Systems

Boundedness of Random Processes

Stability of Markov Chains

Method of Innovations

References

Random Processes of the Statistical Radio Engineering

Power Spectrum of a Stationary Random Process

Wide-Band and Narrow-Band Processes

Random Process with a Discrete Spectrum

Mutual Power Spectrum

Envelope and Phase of a Random Process

Representation of Narrow-Band Processes

Envelope and Phase of a Gaussian Process

Impulse Random Processes

Some Types of Impulse Processes

References

Renewal Theory

Renewal Equation

Renewal Process

Rate of the Convergence

Uniform Theorems

Transient Phenomena

Markov Renewal Equation

References

Branching Processes

Galton-Watson Processes

Bellman-Harris Processes

Markov Branching Processes

Sevastyanov Model

Processes with Several Types of Particles

Jirina Processes