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,…mehr
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 GSummary of the Contents 1. Definitions and General Properties of Random Processes 2. Classification of Random Processes 3. Discrete-Time Markov Chains 4. Main Classes of Constructively Defined Random Processes 5. Random Processes with Independent Increments 6. Processes Associated with a Poisson Process 7. Random Flows of Events 8. Classes of Constructively Defined Random Processes 9. Some Special Classes of Processes 10. Stability of Random Processes 11. Random Processes of the Statistical Radio Engineering 12. Renewal Theory 13. Branching Processes 14. Ergodic Theory and Stationary Processes 15. Markov Processes 16. Statistics for Some Classes of Random Processes 17. Simulation of Random Processes
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 GSummary of the Contents 1. Definitions and General Properties of Random Processes 2. Classification of Random Processes 3. Discrete-Time Markov Chains 4. Main Classes of Constructively Defined Random Processes 5. Random Processes with Independent Increments 6. Processes Associated with a Poisson Process 7. Random Flows of Events 8. Classes of Constructively Defined Random Processes 9. Some Special Classes of Processes 10. Stability of Random Processes 11. Random Processes of the Statistical Radio Engineering 12. Renewal Theory 13. Branching Processes 14. Ergodic Theory and Stationary Processes 15. Markov Processes 16. Statistics for Some Classes of Random Processes 17. Simulation of Random Processes
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