Abstract
Loosely speaking, a point process is a random ‘discrete’ set of points in some Polish space. Thus, one could use a point process to model experiments like throwing grains of sand onto the floor and noting their locations, or pointing an astronomical telescope in a random direction and noting the positions of the stars seen in the field of view. A mathematical example would be the random set of values taken by a finite sequence of random variables. This latter example makes it clear that we may want to generalize the notion of sets to allow a given point to appear more than once. It turns out that there is a nice mathematical way to accommodate the generalization using a certain class of ℤ̄+ -valued measures. The relevant definitions and basic facts are given in the first section. The most important point processes are ‘Poisson point processes’, which are characterized by the property that their intersections with disjoint subsets of the underlying Polish space are independent. These are treated in Sections 3 and 4. An important tool for studying the distributions of point processes is introduced in the fourth section. This tool is needed in the final two sections of the chapter, where various operations on point processes are studied. In particular, the convergence in distribution of point processes is considered in the final section. One nice result from that section is that the Poisson point processes arise as limits of certain naturally defined sequences.
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References
Aldous, David, Probability Approximations via the Poisson Clumping Heuristic, Springer-Verlag, New York, 1989.
Ambartzumian, R. V., Factorization Calculus and Geometric Probability (Encyclopedia of Mathematics and Its Applications, Vol. 33), Cambridge University Press, Cambridge, 1990.
Brémaud, Pierre, Point Processes and Queues: Martingale Dynamics, Springer-Verlag, New York, 1981.
Cox, D. R. and Isham, Valerie, Point Processes, Chapman and Hall, London, 1980.
Daley, D. J. and Vere-Jones, D., An Introduction to the Theory of Point Processes, Springer-Verlag, New York, 1988.
Pranken, Peter and König, Dieter and Arndt, Ursula and Schmidt, Volker, Queues and Point Processes, John Wiley & Sons, Chichester, 1980.
Hall, Peter, Introduction to the Theory of Coverage Processes, John Wiley & Sons, New York, 1988.
Kallenberg, Olav, Random Measures, Akademie-Verlag, Berlin, 1983.
Kingman, J. F. C, Poisson Processes, Clarendon Press, Oxford, 1993.
Matheron, G., Random Sets and Integral Geometry, John Wiley & Sons, New York, 1975.
Matthes, Klaus and Kerstan, Johannes and Mecke, Joseph, Infinitely Divisible Point Processes, John Wiley & Sons, Chichester, 1978.
Molchanov, Ilya S., Limit Theorems for Unions of Random Closed Sets, Springer-Verlag, Berlin, 1993.
Resnick, Sidney I., Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, 1987.
Reiss, R.-D., A Course on Point Processes, Springer-Verlag, New York, 1993.
Santaló, Luis A., Integral Geometry and Geometric Probability (Encyclopedia of Mathematics and Its Applications, Vol. 1), Addison-Wesley, Reading, Massachusetts, 1976.
Solomon, Herbert, Geometric Probability, Society of Industrial and Applied Mathematics, Philadelphia, 1978.
Stoyan, Dietrich and Stoyan, Helga Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics, John Wiley & Sons, Chichester, 1994.
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© 1997 Springer Science+Business Media New York
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Fristedt, B., Gray, L. (1997). Point Processes. In: A Modern Approach to Probability Theory. Probability and its Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-2837-5_29
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DOI: https://doi.org/10.1007/978-1-4899-2837-5_29
Publisher Name: Birkhäuser, Boston, MA
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