# Book:D.R. Cox/The Theory of Stochastic Processes

## D.R. Cox and H.D. Miller: The Theory of Stochastic Processes

Published $1965$, Chapman and Hall.

### Contents

Preface
1 Introduction
1.1 Examples of stochastic processes
1.2 Specification of stochastic processes
1.3 Markov processes
Bibliographic notes and exercises
2 The Random Walk
2.1 Introduction
2.2 The simple random walk
2.3 The general one-dimensional random walk in discrete time
2.4 Further topics
Bibliographic notes and exercises
3 Markov Chains
3.1 Introduction
3.2 A two-state Markov chain
3.3 General definitions and some examples
3.4 The classification of states and the limit theorem
3.5 Closed sets of states
3.6 Irreducible chains and equilibrium distributions
3.7 Branching processes
3.8 Limiting properties of irreducible chains
3.9 Absorption problems
3.10 Non-negative square matrices
3.11 Finite Markov chains
3.12 Further topics
3.13 Appendix on power series with non-negative coefficients
Bibliographic notes and exercises
4 Markov Processes with Discrete States in Continuous Time
4.1 The Poisson process
4.2 Generalizations of the Poisson process
4.3 Some simple processes of the birth-death type
4.4 Equilibrium distributions
4.5 General formulation
4.6 Some miscellaneous topics
Bibliographic notes and exercises
5 Markov Processes in Continuous Time with Continuous State Space
5.1 Introduction
5.2 Continuous limit of the simple random walk: the Wiener process
5.3 The diffusion equations for the Wiener process
5.4 First passage problems for the Wiener process
5.5 Continuous limits of more general discrete processes
5.6 The Kolmogorov equations
5.7 Boundary conditions for homogeneous diffusion processes
5.8 The Ornstein-Uhlenbeck process
5.9 Transformations of the Wiener process
5.10 First passage times for homogeneous diffusion processes
5.11 Approximations to discrete processes by means of diffusion processes
5.12 Continuous and jump transitions
5.13 Processes with independent increments
5.14 Multidimensional processes
Bibliographic notes and exercises
6 Non-Markovian Processes
6.1 Introduction
6.2 The device of stages
6.3 Supplementary variables
6.4 Imbedded Markov process
Bibliographic notes and exercises
7 Stationary Processes: Time Domain
7.1 Introduction
7.2 Some definitions and special processes
7.3 Some general results about stationary processes
7.4 Processes in continuous time
7.5 Prediction theory
Bibliographic notes and exercises
8 Stationary Processes: Frequency Domain
8.1 Introduction
8.2 The spectral representation
8.3 Linear operations on stationary processes
8.4 Derivation of the spectral representation
8.5 Prediction and filtering theory
8.6 Multivariate processes
Bibliographic notes and exercises
9 Point Processes
9.1 Introduction page
9.2 The renewal process
9.3 Renewal processes with more than one type of interval
9.4 Stationary point processes
9.5 Operations on point processes
9.6 Real-valued processes associated with a point process
Bibliographic notes and exercises
Appendix 1 Table of exponentially distributed random quantities
Appendix 2 Bibliography
Author index
Subject index