Stochastic Processes

The following sections are included:

• Contents

• PREFACE

• ABBREVIATIONS AND NOTATIONS

The following sections are included:

• Introduction

• The Exponential Density

• The Gamma Density

• The Beta Density

• The Uniform Density

• The Cauchy Density

• The Normal Density in One Dimension
  • Convolution Property

• The Normal Density in n Dimensions

• Infinitely Divisible Distributions
  • Examples

• Stable Distributions
  • Examples

• Problems for Solution

The following sections are included:

• Introduction

• Analytic Definition

• Definition in Terms of Finite-Dimensional Distributions

• Moments of Stochastic Processes

• Some Problems in Stochastic Processes

• Probability Models

• Comments on the Definition of a Stochastic Process

• Further Reading

• Books of Historical Interest

The following sections are included:

• Stationary Processes

• Processes with Stationary Independent Increments

• Markov Processes

• Problems for Solution

The following sections are included:

• Examples of Real Stationary Processes

• The General Case

• A Second Order Calculus for Stationary Processes

• Time Series Models

• Mean Square Convergence

• Problems for Solution

• Further Reading

The following sections are included:

• The Brownian Motion
  • Historical Remarks
  • Introduction
  • Properties of the Brownian Motion

• The Poisson Process
  • Introduction
  • Properties of the Poisson Process
  • The Compound Poisson Process

• Lévy Processes

• The Gaussian Process
  • Application to Brownian Storage Models

• The Inverse Gaussian Process

• The Randomized Bernoulli Random Walk
  • Application to the Simple Queue

• Lévy Processes: Further Properties

• Problems for Solution

• Further Reading

The following sections are included:

• Renewal Processes: Introduction
  • Physical Interpretation

• The Renewal-Counting Processes {N(t)}

• Renewal Theorems

• The Age and the Remaining Lifetime

• The Stationary Renewal Process

• The Case of the Infinite Mean

• The Random Walk on the Real Line: Introduction

• The Maximum and Minimum Functionals

• Ladder Processes

• Limit Theorems for M_n

• Problems for Solution

• Further Reading

The following sections are included:

• Introduction and Examples

• Some Terminology

• Martingales Relative to a Sigma-Field

• Decision Functions; Optional Stopping

• Submartingales and Supermartingales

• Optional Skipping and Sampling Theorems

• Application to Random Walks

• Convergence Properties

• The Concept of Fairness

• Problems for Solution

• Further Reading

The following sections are included:

• Introduction

• The Problem of Extinction

• The Extinction Time and the Total Progeny

• The Supercritical Case

• Estimation

• Problems for Solution

• Further Reading

The following sections are included:

• Introduction

• Discrete Time Regenerative Phenomena

• Subordination of Renewal Counting Processes

• The Simple Random Walk in D Dimensions

• The Bernoulli Random Walk

• Ladder Sets of Random Walks on the Real Line

• Further Examples of Recurrent Phenomena

• Regenerative Phenomena in Continuous Time

• Stable Regenerative Phenomena

• Problems for Solution

• Further Reading

The following sections are included:

• Introduction

• Discrete Time Markov Chains

• Examples of Finite Markov Chains
  • Markov Trials
  • The Bernoulli–Laplace Diffusion Model

• The Limit Distribution of Finite Markov Chains

• Classification of States. Limit Theorems

• Closed Sets. Irreducible Chains

• Stationary Distributions

• Examples of Infinite Markov Chains
  • The Branching Process as a Markov Chain
  • The Queueing System GI/M/1

• Continuous Time Markov Chains

• Examples of Continuous Time Markov Chains
  • The Poisson Process as a Markov Chain
  • The Pure Birth Process
  • The Pure Death Process
  • The Birth and Death Process

• Models for Population Growth
  • Some Deterministic Models
  • Stochastic Models
  • The Yule–Furry Model
  • The Feller–Arley Model
  • The Kendall Model
  • The Differential Equation

• Problems for Solution
  • Finite Markov Chains
  • Infinite Markov Chains
  • Continuous Time Markov Chains
  • Models for Population Growth

The following sections are included:

• Appendix A: Tauberian Theorems

• Appendix B: Some Asymptotic Relations

• INDEX