Elements of Stochastic Processes.

Format:

Book

Author/Creator:

TANG, Jiashan.

Series:

Textbooks for Tomorrow's Scientists Series:Textbooks for Tomorrow's Scientists Introduces Advanced Textbooks on a Wide Range of Topics in STM Subject Areas. These Books Offer Students, Researchers, Faculty and Professionals the Resources They Need to Lear

Language:

English

Physical Description:

1 online resource (374 pages)

Edition:

1st ed.

Place of Publication:

Les Ulis : EDP Sciences, 2025.

Summary:

This book introduces the fundamental elements of stochastic processes, covering basic concepts, core knowledge, and essential methods.

Contents:

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 σ-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.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

2-7598-3919-2

9782759839193

OCLC:

1545255573