Elements of Stochastic Processes
1st Edition
Published on 14. October 2025
374 pages
978-2-7598-3919-3 (ISBN)
Description
This book introduces the fundamental elements of stochastic processes, covering basic concepts, core knowledge, and essential methods. It is organized into seven chapters, which include: the foundations of probability, concepts of stochastic processes, Poisson processes, discrete-time Markov chains, continuous-time Markov chains, simple Markovian queueing models, and stationary processes. Each chapter features figures, examples, exercises, and either solutions or guidance to support learning. Written from a non-measure-theoretic perspective, the book is accessible and highly readable, making it especially suitable for self-study.The intended audience includes senior undergraduate and junior graduate students, as well as teachers in fields such as engineering, economics, management science, operations research, and applied mathematics. It also serves as a valuable reference for researchers and professionals with an interest in stochastic processes.
More details
Person
Contributor: Jiashan TANG Jiashan TANG is full professor of mathematics and director of the Stochastics Research Center at Nanjing University of Posts and Telecommunications (NUPT). He received his Ph.D. in mathematics from Beijing Normal University in 1997 and was a Postdoctoral Fellow in the School of Mathematics and Statistics at Carleton University (2002-2003). His research interests include probability theory, applied stochastic processes, queueing theory, signal and information processing, applied statistics, and data science.
Content
- Cover-1
- Cover-2
- Elements of Stochastic Processes
- Preface
- Contents
- Chapter 1 Basis of probability
- 1.1 Random events
- 1.1.1 Random experiments
- 1.1.2 Random events
- 1.1.3 Calculation among random events
- 1.1.4 s-algebras
- 1.2 Probability
- 1.2.1 What is probability
- 1.2.2 Properties of probability
- 1.2.3 Two fundamental probability models
- 1.2.4 Conditional probability and three important formulas
- 1.2.5 Independence of events
- 1.3 Random variables
- 1.3.1 Definition
- 1.3.2 Cumulative distribution functions
- 1.3.3 Classification
- 1.3.4 Multiple-dimensional random variables
- 1.3.5 Distributions of functions of random variables
- 1.3.6 Conditional distributions and independences
- 1.3.7 Order statistics
- 1.4 Numerical characteristics
- 1.4.1 Definition of mathematical expectations
- 1.4.2 Properties
- 1.4.3 Other numerical characteristics of random variables
- 1.4.4 Conditional mathematical expectations
- 1.5 Limiting theorems
- 1.5.1 Convergence of random variables
- 1.5.2 Law of large numbers
- 1.5.3 Central limit theorems
- Exercise 1
- Answers or tips for Exercise 1
- Chapter 2 Stochastic processes
- 2.1 Definition and classification
- 2.1.1 Definition
- 2.1.2 Classification
- 2.1.3 Examples
- 2.2 Statistical laws of stochastic processes
- 2.2.1 Finite dimensional distribution functions
- 2.2.2 Kolmogorov's theorem
- 2.3 Measurements of stochastic processes
- 2.3.1 Measurements for one stochastic process
- 2.3.2 Measurements for two stochastic processes
- 2.4 Further comments
- Exercise 2
- Answers or tips for Exercise 2
- Chapter 3 Poisson processes
- 3.1 Definition and measurements
- 3.1.1 Definition
- 3.1.2 Measurements
- 3.2 Waiting times and interarrival times
- 3.2.1 Waiting times
- 3.2.2 Interarrival times
- 3.3 Conditional distributions
- 3.4 Extensions of Poisson processes
- 3.4.1 Non-homogeneous Poisson processes
- 3.4.2 Compound Poisson processes
- 3.4.3 Conditional Poisson processes
- 3.4.4 Renewal processes
- Exercise 3
- Answers or tips for Exercise 3
- Chapter 4 Discrete-time Markov chains
- 4.1 Definition of discrete-time Markov chains
- 4.1.1 Definition
- 4.1.2 Chapman-Kolmogorov equations
- 4.2 Finite-dimensional distributions
- 4.3 Properties of a single state
- 4.3.1 Periods
- 4.3.2 Transience and recurrence
- 4.4 Decomposition of state space
- 4.4.1 Equivalence relation
- 4.4.2 Decomposition of state space
- 4.5 Asymptotic behaviors of transition probabilities Pij(n)
- 4.5.1 Case one: state j is transient or null-recurrent
- 4.5.2 Case two: state j is positive-recurrent
- 4.6 Stationary distributions
- 4.6.1 Definition of stationary distributions
- 4.6.2 How many stationary distributions a Markov chain may have?
- 4.6.3 Rates of convergence to stationary distributions
- 4.6.4 Stationary distributions of a censored Markov chain
- 4.6.5 Quasi-stationary distributions
- 4.7 Reversible Markov chains
- Exercise 4
- Answers or tips for Exercise 4
- Chapter 5 Continuous-time Markov chains
- 5.1 Definition of continuous-time Markov chains
- 5.1.1 Definition
- 5.1.2 Chapman-Kolmogorov equations
- 5.2 Finite-dimensional distributions
- 5.3 Q-matrices
- 5.4 Kolmogorov differential equations
- 5.5 Asymptotic behaviors
- 5.5.1 Transience and recurrence
- 5.5.2 Limiting results
- 5.5.3 Stationary distributions
- 5.6 Birth and death processes
- Exercise 5
- Answers or tips for Exercise 5
- Chapter 6 Simple Markovian queueing models
- 6.1 Terminology and notation
- 6.2 Little's law and PASTA property
- 6.2.1 Little's law
- 6.2.2 PASTA property
- 6.3 M/M/1 queueing model
- 6.3.1 Stationary distribution of queue length
- 6.3.2 Distributions of sojourn times and waiting times
- 6.3.3 Busy period distribution
- 6.3.4 Departure process
- 6.4 M/M/n and state dependent M/M/1 queueing model
- 6.4.1 M/M/n queueing model
- 6.4.2 State dependent M/M/1 queueing model
- 6.5 Mx /M/1 queueing model
- 6.6 M/G/1 queueing model
- 6.6.1 Embedded Markov chain
- 6.6.2 M/Er/1 queueing model
- Exercise 6
- Answers or tips for Exercise
- Chapter 7 Stationary processes
- 7 .1 Definition
- 7.1.1 Strict-sense stationary processes
- 7.1.2 Wide-sense stationary processes
- 7.2 Analytic properties of wide-sense stationary processes
- 7.3 Correlation functions and their spectra
- 7.3.1 Properties of correlation functions
- 7.3.2 Spectral density functions
- 7.3.3 Properties of spectral density functions
- 7.3.4 Continuous-time white noise processes
- 7.4 Ergodicity
- 7.5 Passing through a linear time-invariant system
- Exercise 7
- Answers or tips for Exercise 7
- References
- Index
