Point Processes

Point processes are random systems consisting of point events occurring in space or time. The object of this book is to develop the applied mathemathics of point processes at a level which will make the ideas accessible both to the research worker and the postgraduate student in probability and statistics.

Klappentext

There has been much recent research on the theory of point processes, i.e., on random systems consisting of point events occurring in space or time. Applications range from emissions from a radioactive source, occurrences of accidents or machine breakdowns, or of electrical impluses along nerve fibres, to repetitive point events in an individual's medical or social history. Sometimes the point events occur in space rather than time and the application here raneg from statistical physics to geography. The object of this book is to develop the applied mathemathics of point processes at a level which will make the ideas accessible both to the research worker and the postgraduate student in probability and statistics and also to the mathemathically inclined individual in another field interested in using ideas and results. A thorough knowledge of the key notions of elementary probability theory is required to understand the book, but specialised "pure mathematical" coniderations have been avoided.

Inhalt

Preface

1 Introduction

• Preliminary remarks

• Four illustrative examples

• Poisson processes

• Renewal processes

• Linear self-exciting processes

• Doubly stochastic Poisson processes

• The specification and properties of point processes

• Some generalizations

• Multiple occurrences

• Multivariate processes

• Marked processes

• Spatial and multidimensional processes

• Discrete time processes

Bibliographic notes, 1

Further results and exercises, 1

• Theoretical framework

  • Some basic definitions

  • Stationarity

  • Orderliness

  • Palm distributions

  • Moments

  • Spectral properties

  • The probability generating functional

  • Multivariate and multidimensional processes

Bibliographic notes, 2

Further results and exercises, 2

• Special Models

  • Poisson processes

• Introduction

• Non-stationary Poisson processes

• Compound Poisson processes

• Cluster processes

• Doubly stochastic Poisson processes

• Renewal processes and generalizations

  Introduction

  Semi-Markov processes

  Moran's process with pairwise dependent intervals

  Processes with Markov-depended inte