随机过程是建立各种类型的大量随机变量现象模型的必要依据，作为应用概率方向的一个工具，书中将离散空间，Markov链，更新理论，点过程，分支过程，随机游程，Brownian运动，这些论题都是生动地展现给读者。《随机过程探究》表述灵活，大量的例子，练习和应用，并有的计算机程序作支持，使得内容的立体感增强，易于理解，可以作为应用科学领域不同层次水平学生的对随机过程的入门教程。每章末附有大量的补充练习。

Preface

CHAPTER 1.PRELIMINARIES" DISCRETE INDEX SETS AND/OR DISCRETE STATE SPACES

1.1.Non-negative integer valued random variables

1.2.Convolution

1.3.Generating functions

 1.3.1.Differentiation of generating functions

 1.3.2.Generating functions and moments

 1.3.3.Generating functions and convolution

 1.3.4.Generating functions, compounding and random sums

1.4.The simple branching process

1.5.Limit distributions and the continuity theorem

 1.5.1.The law of rare events

1.6.The simple random walk

1.7.The distribution of a process*

1.8.Stopping times*

 1.8.1.Wald's identity

 1.8.2.Splitting an iid sequence at a stopping time ~

Exercises for Chapter 1

CHAPTER 2.MARKOV CHAINS

2.1.Construction and first properties

2.2.Examples

2.3.Higher order transition probabilities

2.4.Decomposition of the state space

2.5.The dissection principle

2.6.Transience and recurrence

2.7.Periodicity

2.8.Solidarity properties

2.9.Examples

2.10.Canonical decomposition

2.11.Absorption probabilities

2.12.Invariant measures and stationary distributions

 2.12.1.Time averages

2.13.Limit distributions

 2.13.1 More on null recurrence and transience*

2.14.Computation of the stationary distribution

2.15.Classification techniques

Exercises for Chapter 2

CHAPTER 3.RENEWAL THEORY

3.1.Basics

3.2.Analytic interlude

 3.2.1.Integration

 3.2.2.Convolution

 3.2.3.Laplace transforms

3.3.Counting renewals

3.4.Renewal reward processes

3.5.The renewal equation

 3.5.1.Risk processes*

3.6.The Poisson process as a renewal process

3.7.Informal discussion of renewal limit theorems; regenerative processes

 3.7.1 An informal discussion of regenerative processes

3.8.Discrete renewal theory

3,9.Stationary renewal processes* .

3.10.Blackwell and key renewal theorems* .

 3.10.1.Direct Riemann integrability*

 3.10.2.Equivalent forms of the renewal theorems*

 3.10.3.Proof of the renewal theorem*

3.11.Improper renewal equations

3.12.More regenerative processes*

 3.12.1.Definitions and examples*

 3.12.2.The renewal equation and Smith's theorem*

 3.12.3.Queueing examples

Exercises for Chapter 3

CHAPTER 4.POINT PROCESSES

4.1.Basics

4.2.The Poisson process

4.3.Transforming Poisson processes

 4.3.1.Max-stable and stable random variables*

4.4.More transformation theory; marking and thinning

4.5.The order statistic property

4.6.Variants of the Poisson process

4.7.Technical basics*

 4.7.1.The Laplace functional*

4.8.More on the Poisson process*

4.9.A general construction of the Poisson process; a simple derivation of the order statistic property*

4.10.More transformation theory; location dependent thinning*

4.11.Records*

Exercises for Chapter 4

CHAPTER 5.CONTINUOUS TIME MARKOV CHAINS

5.1.Defiuitions and construction

5.2.Stability and explosions

 5.2.1.The Markov property* .

5.3.Dissection

 5.3.1.More detail on dissection*

5.4.The backward equation and the generator matrix

5.5.Stationary and limiting distributions

 5.5.1.More on invariant measures*

5.6.Laplace transform methods

5.7.Calculations and examples

 5.7.1.Queueing networks

5.8.Time dependent solutions*

5.9.Reversibility

5.10.Uniformizability

5.11.The linear birth process as a point process

Exercises for Chapter 5

CHAPTER 6.BROWNIAN MOTION

6.1.Introduction

6.2.Preliminaries

6.3.Construction of Brownian motion*

6.4.Simple properties of standard Brownian motion

6.5.The reflection principle and the distribution of the maximum

6.6.The strong independent increment property and reflection*

6.7.Escape from a strip

6.8.Brownian motion with drift

6.9.Heavy traffic approximations in queueing theory

6.10.The Brownian bridge and the Kolmogorov--Smirnov statistic.

6.11.Path properties*

6.12.Quadratic variation

6.13.Khintchine's law of the iterated logarithm for Brownian motion

Exercises for Chapter 6

CHAPTER 7.THE GENERAL RANDOM WALK*

7.1.Stopping times

7.2.Global properties

7.3.Prelude to Wiener-Hopf: Probabilistic interpretations of transforms

7.4.Dual pairs of stopping times

7.5.Wiener-Hopf decompositions

7.6.Consequences of the Wiener-Hopf factorization

7.7.The maximum of a random walk

7.8.Random walks and the G/G/1 queue

 7.8.1.Exponential right tail

 7.8.2.Application to G/M/1 queueing model

 7.8.3.Exponential left tail

 7.8.4.The M/G/1 queue

 7.8.5.Queue lengths

References

Index