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

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## D.R. Cox and H.D. Miller:

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

Published $\text {1965}$, **Chapman and Hall**.

### Subject Matter

### 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