Models of Random Processes: A Handbook for Mathematicians and Engineers

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.

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