# Poisson Point Process

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A Poisson Point Process is a stochastic point process for a … (in which stochastic events occur continuously and independently of one another).

## References

### 2018

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Point_process#Poisson_point_process Retrieved:2018-3-1.
• The simplest and most ubiquitous example of a point process is the Poisson point process, which is a spatial generalisation of the Poisson process. A Poisson (counting) process on the line can be characterised by two properties : the number of points (or events) in disjoint intervals are independent and have a Poisson distribution. A Poisson point process can also be defined using these two properties. Namely, we say that a point process $\displaystyle{ \xi }$ is a Poisson point process if the following two conditions hold

1) $\displaystyle{ \xi(B_1),\ldots,\xi(B_n) }$ are independent for disjoint subsets $\displaystyle{ B_1,\ldots,B_n. }$ 2) For any bounded subset $\displaystyle{ B }$ , $\displaystyle{ \xi(B) }$ has a Poisson distribution with parameter $\displaystyle{ \lambda \|B\|, }$ where $\displaystyle{ \|\cdot\| }$ denotes the Lebesgue measure.

The two conditions can be combined together and written as follows : For any disjoint bounded subsets $\displaystyle{ B_1,\ldots,B_n }$ and non-negative integers $\displaystyle{ k_1,\ldots,k_n }$ we have that : $\displaystyle{ \Pr[\xi(B_i) = k_i, 1 \leq i \leq n] = \prod_i e^{-\lambda \|B_i\|}\frac{(\lambda \|B_i\|)^{k_i}}{k_i!}. }$ The constant $\displaystyle{ \lambda }$ is called the intensity of the Poisson point process. Note that the Poisson point process is characterised by the single parameter $\displaystyle{ \lambda. }$ It is a simple, stationary point process.

To be more specific one calls the above point process, a homogeneous Poisson point process. An inhomogeneous Poisson process is defined as above but by replacing $\displaystyle{ \lambda \|B\| }$ with $\displaystyle{ \stackrel{}{} \int_B\lambda(x) \, dx }$ where $\displaystyle{ \lambda }$ is a non-negative function on $\displaystyle{ \mathbb{R}^d. }$

### 2016a

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Poisson_point_process Retrieved:2016-7-18.
• In probability, statistics and related fields, a Poisson point process or Poisson process (also called a Poisson random measure, Poisson random point field or Poisson point field) is a type of random mathematical object that consists of points randomly located on a mathematical space.[1] The process has convenient mathematical properties,[2] which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,[3] biology,[4] ecology,[5] geology,[6] physics,[7] image processing,[8] and telecommunications.[9] [10]

The Poisson point process is often defined on the real line. For example, in queueing theory [11] it is used to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in time. In the plane, the point process, also known as a spatial Poisson process,[12] may represent scattered objects such as transmitters in a wireless network, [13] [14] [15] particles colliding into a detector, or trees in a forest. In this setting, the process is often used in mathematical models and in the related fields of spatial point processes, [16] stochastic geometry, spatial statistics [17] and continuum percolation theory.[18] In more abstract spaces, the Poisson point process serves as an object of mathematical study in its own right.

In all settings, the Poisson point process has the property that each point is stochastically independent to all the other points in the process, which is why it is sometimes called a purely or completely random process. Despite its wide use as a stochastic model of phenomena representable as points, the inherent nature of the process implies that it does not adequately describe phenomena in which there is sufficiently strong interaction between the points. This has sometimes led to the overuse of the point process in mathematical models, and has inspired other point processes, some of which are constructed via the Poisson point process, that seek to capture this interaction.

The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. [19] [20] It's named owing to the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is directly related to the Poisson distribution. The process was discovered independently in several different settings.

The process is defined with a single non-negative mathematical object, which, depending on the context, may be a constant, an integrable function or, in more general settings, a Radon measure. If this object is a constant, then the resulting process is called a homogeneous or stationary Poisson point process. Otherwise, the parameter depends on its location in the underlying space, which leads to the inhomogeneous or nonhomogeneous Poisson point process. The word point is often omitted, but there are other Poisson processes of objects, which, instead of points, consist of more complicated mathematical objects such as lines and polygons, and such processes can be based on the Poisson point process.

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2. J. F. C. Kingman. Poisson processes, volume 3. Oxford university press, 1992.
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4. H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. Journal of mathematical biology, 26(3):263-298, 1988.
5. H. Thompson. Spatial point processes, with applications to ecology. Biometrika, 42(1/2):102-115, 1955.
6. C. B. Connor and B. E. Hill. Three nonhomogeneous poisson models for the probability of basaltic volcanism: application to the yucca mountain region, nevada. Journal of Geophysical Research: Solid Earth (1978-2012), 100(B6):10107-10125, 1995.
7. J. D. Scargle. Studies in astronomical time series analysis. v. bayesian blocks, a new method to analyze structure in photon counting data. The Astrophysical Journal, 504(1):405, 1998.
8. M. Bertero, P. Boccacci, G. Desidera, and G. Vicidomini. Image deblurring with poisson data: from cells to galaxies. Inverse Problems, 25(12):123006, 2009.
9. F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks, Volume II- Applications, volume 4, No 1-2 of Foundations and Trends in Networking. NoW Publishers, 2009.
10. M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti. Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE JSAC, 27(7):1029-1046, september 2009.
11. L. Kleinrock. Theory, volume 1, Queueing systems. Wiley-interscience, 1975.
12. A. Baddeley, I. Bárány, and R. Schneider. Spatial point processes and their applications. Stochastic Geometry: Lectures given at the CIME Summer School held in Martina Franca, Italy, September 13–18, 2004, pages 1-75, 2007.
13. J. G. Andrews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Weber. A primer on spatial modeling and analysis in wireless networks. Communications Magazine, IEEE, 48(11):156-163, 2010.
14. F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks, Volume I - Theory, volume 3, No 3-4 of Foundations and Trends in Networking. NoW Publishers, 2009.
15. M. Haenggi. Stochastic geometry for wireless networks. Cambridge University Press, 2012.
16. D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods, Springer, New York, second edition, 2003.
17. J. Møller and R. P. Waagepetersen. Statistical inference and simulation for spatial point processes. CRC Press, 2003.
18. R. Meester and R. Roy. Continuum percolation, volume 119 of cambridge tracts in mathematics, 1996.
19. D. Stirzaker. Advice to hedgehogs, or, constants can vary. The Mathematical Gazette, 84(500):197-210, 2000.sing

20. P. Guttorp and T. L. Thorarinsdottir. What happened to discrete chaos, the quenouille process, and the sharp markov property? some history of stochastic point processes. International Statistical Review, 80(2):253-268, 2012.

### 2016b

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Poisson_point_process#Overview_of_definitions Retrieved:2016-7-18.
• The Poisson point process is one of the most studied point processes, in both the field of probability and in more applied disciplines concerning random phenomena, due to its convenient properties as a mathematical model as well as being mathematically interesting. Depending on the setting, the process has several equivalent definitions [1] as well definitions of varying generality owing to its many applications and characterizations. It may be defined, studied and used in one dimension (on the real line) where it can be interpreted as a counting process or part of a queueing model; [2] in higher dimensions such as the plane where it plays a role in stochastic geometry and spatial statistics; [3] or on more abstract mathematical spaces.[4] Consequently, the notation, terminology and level of mathematical rigour used to define and study the Poisson point process and points processes in general vary according to the context. Despite its different forms and varying generality, the Poisson point process has two key properties.
1. H. C. Tijms. A first course in stochastic models. Wiley. com, 2003.
2. S. Ross. Stochastic processes. Wiley series in probability and statistics: Probability and statistics. Wiley, 1996.
3. A. Baddeley. A crash course in stochastic geometry. Stochastic Geometry: Likelihood and Computation Eds OE Barndorff-Nielsen, WS Kendall, HNN van Lieshout (London: Chapman and Hall) pp, pages 1-35, 1999.
4. D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure, Springer, New York, second edition, 2008.