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{{Short description|Type of random mathematical object}}
{{Use dmy dates|date=July 2021}}
{{Infobox probability distribution
[[File:Poisson process.svg|thumb|alt=Poisson point process|A visual depiction of a Poisson point process starting from 0, in which increments occur continuously and independently at rate ''λ''.]]▼
|name=Poisson Process
In [[probability]], statistics and related fields, a '''Poisson point process''' is a type of [[random]] [[mathematical object]] that consists of [[Point (geometry)|points]] randomly located on a [[mathematical space]].<ref name="ChiuStoyan2013">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3}}</ref> The Poisson point process is often called simply the '''Poisson process''', but it is also called a '''Poisson random measure''', '''Poisson random point field''' or '''Poisson point field'''. This [[point process]] has convenient mathematical properties,<ref name="Kingman1992">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2}}</ref> which has led to its being frequently defined in [[Euclidean space]] and used as a [[mathematical model]] for seemingly random processes in numerous disciplines such as [[astronomy]],<ref name="babu1996spatial">G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311–326, 1996.</ref> [[biology]],<ref name="othmer1988models">H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263–298, 1988.</ref> ecology,<ref name="thompson1955spatial">H. Thompson. Spatial point processes, with applications to ecology. ''Biometrika'', 42(1/2):102–115, 1955.</ref> geology,<ref name="connor1995three">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.</ref> [[seismology]],<ref>{{Cite journal|last1=Gardner|first1=J. K.|last2=Knopoff|first2=L.|date=1974|title=Is the sequence of earthquakes in Southern California, with aftershocks removed, Poissonian?|url=https://pubs.geoscienceworld.org/ssa/bssa/article-abstract/64/5/1363/117341/is-the-sequence-of-earthquakes-in-southern|journal=Bulletin of the Seismological Society of America|volume=64|issue=5 |pages=1363–1367|doi=10.1785/BSSA0640051363 |s2cid=131035597 }}</ref> [[physics]],<ref name="scargle1998studies">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.</ref> economics,<ref name="AghionHowitt1992">P. Aghion and P. Howitt. A Model of Growth through Creative Destruction. ''Econometrica'', 60(2). 323–351, 1992.</ref> [[image processing]],<ref name="bertero2009image">M. Bertero, P. Boccacci, G. Desidera, and G. Vicidomini. Image deblurring with poisson data: from cells to galaxies. ''Inverse Problems'', 25(12):123006, 2009.</ref><ref>{{cite web | url=https://caseymuratori.com/blog_0010 | title=The Color of Noise }}</ref> and telecommunications.<ref name="baccelli2009stochastic2">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.</ref><ref name="Haenggi2009">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.</ref>▼
|pdf_image=[[File:Poisson Process.png|325px]]
|mean=<math>a_{0, t} = \int_{0}^{t} \lambda(\alpha) d\alpha</math>
|variance=<math>a_{0, t} + (a_{0, t})^2 - (a_{0, t})^2 = a_{0, t}</math> <br>
since <math>R_x(t_1,t_2) = a_{0, min(t_1,t_2)} + a_{0, t_1} a_{0, t_2}</math>
where for <math>E\{X^2\} = R_x(t,t) = a_{0, t} + (a_{0, t})^2</math>
|type=multivariate
}}
▲[[File:Poisson process.svg|thumb|alt=Poisson point process|A visual depiction of a Poisson point process starting
▲
The Poisson point process is often defined on the [[real line]], where it can be considered as a [[stochastic process]]. In this setting, it is used, for example, in [[queueing theory]]<ref name="Kleinrock1976">{{cite book|author=Leonard Kleinrock|title=Queueing Systems: Theory|url=https://archive.org/details/queueingsystems01klei|url-access=registration|year=1976|publisher=Wiley|isbn=978-0-471-49110-1}}</ref> to model random events, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes, distributed in time. In the [[Plane (geometry)|plane]], the point process, also known as a '''spatial Poisson process''',<ref name="BaddeleyBárány2006page10">{{cite book|author1=A. Baddeley|author2=I. Bárány|author3=R. Schneider|title=Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004|url=https://books.google.com/books?id=X-m5BQAAQBAJ|date=26 October 2006|publisher=Springer|isbn=978-3-540-38175-4|page=10}}</ref> can represent the locations of scattered objects such as transmitters in a [[wireless network]],<ref name="baccelli2009stochastic2"/><ref name="andrews2010primer">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.</ref><ref name="baccelli2009stochastic1">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.</ref><ref name="Haenggi2013">{{cite book|author=Martin Haenggi|title=Stochastic Geometry for Wireless Networks|url=https://books.google.com/books?id=CLtDhblwWEgC|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01469-5}}</ref> [[particles]] colliding into a detector, or trees in a forest.<ref name="ChiuStoyan2013page51"/> In this setting, the process is often used in mathematical models and in the related fields of spatial point processes,<ref name="BaddeleyBárány2006">{{cite book|author1=A. Baddeley|author2=I. Bárány|author3=R. Schneider|title=Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004|url=https://books.google.com/books?id=X-m5BQAAQBAJ|date=26 October 2006|publisher=Springer|isbn=978-3-540-38175-4}}</ref> [[stochastic geometry]],<ref name="ChiuStoyan2013"/> [[spatial statistics]]<ref name="BaddeleyBárány2006"/><ref name="MollerWaagepetersen2003">{{cite book|author1=Jesper Moller|author2=Rasmus Plenge Waagepetersen|title=Statistical Inference and Simulation for Spatial Point Processes|url=https://books.google.com/books?id=dBNOHvElXZ4C|date=25 September 2003|publisher=CRC Press|isbn=978-0-203-49693-0}}</ref> and [[continuum percolation theory]].<ref name="meester1996continuum">R. Meester and R. Roy. Continuum percolation, volume 119 of cambridge tracts in mathematics, 1996.</ref> The Poisson point process can be defined on more [[Abstraction (mathematics)|abstract]] spaces. Beyond applications, the Poisson point process is an object of mathematical study in its own right.<ref name="Kingman1992"/> 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.{{sfnp|Daley|Vere-Jones|2003|page=27}} 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 where there is sufficiently strong interaction between the points. This has inspired the proposal of other point processes, some of which are constructed with the Poisson point process, that seek to capture such interaction.<ref name="ChiuStoyan2013page35">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=35–36}}</ref>▼
▲The Poisson point process is often defined on the
The point process depends on a single mathematical object, which, depending on the context, may be a [[Constant (mathematics)|constant]], a [[locally integrable function]] or, in more general settings, a [[Radon measure]].<ref name="ChiuStoyan2013page41and51">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41 and 51 }}</ref> In the first case, the constant, known as the '''rate''' or '''intensity''', is the average [[density]] of the points in the Poisson process located in some region of space. The resulting point process is called a '''homogeneous''' or '''stationary Poisson point process'''.<ref name="ChiuStoyan2013page41">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41–42}}</ref> In the second case, the point process is called an '''inhomogeneous''' or '''nonhomogeneous''' '''Poisson point process''', and the average density of points depend on the ___location of the underlying space of the Poisson point process.{{sfnp|Daley|Vere-Jones|2003|page=22}} The word ''point'' is often omitted,<ref name="Kingman1992" /> but there are other ''Poisson processes'' of objects, which, instead of points, consist of more complicated mathematical objects such as [[line (geometry)|line]]s and [[polygon]]s, and such processes can be based on the Poisson point process.<ref name="Kingman1992page73to76">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|pages=73–76}}</ref> Both the homogeneous and nonhomogeneous Poisson point processes are particular cases of the [[generalized renewal process]].▼
▲The point process depends on a single mathematical object, which, depending on the context, may be a [[Constant (mathematics)|constant]], a [[locally integrable function]] or, in more general settings, a [[Radon measure]].<ref name="ChiuStoyan2013page41and51">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41 and 51 }}</ref> In the first case, the constant, known as the '''rate''' or '''intensity''', is the average [[density]] of the points in the Poisson process located in some region of space. The resulting point process is called a '''homogeneous''' or '''stationary Poisson point process'''.<ref name="ChiuStoyan2013page41">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41–42}}</ref> In the second case, the point process is called an '''inhomogeneous''' or '''nonhomogeneous''' '''Poisson point process''', and the average density of points depend on the ___location of the underlying space of the Poisson point process.{{sfnp|Daley|Vere-Jones|2003|page=22}} The word ''point'' is often omitted,<ref name="Kingman1992">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2}}</ref> but there are other ''Poisson processes'' of objects, which, instead of points, consist of more complicated mathematical objects such as [[line (geometry)|line]]s and [[polygon]]s, and such processes can be based on the Poisson point process.<ref name="Kingman1992page73to76">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|pages=73–76}}</ref> Both the homogeneous and nonhomogeneous Poisson point processes are particular cases of the [[generalized renewal process]].
{{toclimit|limit=3}}
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Depending on the setting, the process has several equivalent definitions<ref name="Tijms2003page1">{{cite book|author=H. C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=RK9yFrNxom8C|date=18 April 2003|publisher=John Wiley & Sons|isbn=978-0-471-49880-3|pages=1–2}}</ref> as well as definitions of varying generality owing to its many applications and characterizations.{{sfnp|Daley|Vere-Jones|2003|pages=26–37}} The Poisson point process can be defined, studied and used in one dimension, for example, on the real line, where it can be interpreted as a counting process or part of a queueing model;<ref name="Tijms2003page1and9">{{cite book|author=H. C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=RK9yFrNxom8C|date=18 April 2003|publisher=John Wiley & Sons|isbn=978-0-471-49880-3|pages=1 and 9}}</ref><ref name="Ross1996page59">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|pages=59–60}}</ref> in higher dimensions such as the plane where it plays a role in [[stochastic geometry]]<ref name="ChiuStoyan2013"/> and [[spatial statistics]];<ref name="baddeley1999crash">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)'', pages 1–35, 1999.</ref> or on more general mathematical spaces.<ref name="DaleyVere-Jones2007page1">{{cite book|author1=D.J. Daley|author2=David Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure|url=https://books.google.com/books?id=nPENXKw5kwcC|date=12 November 2007|publisher=Springer Science & Business Media|isbn=978-0-387-21337-8|pages=1–2}}</ref> 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.<ref name="ChiuStoyan2013page110to111">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=110–111 }}</ref>
Despite all this, the Poisson point process has two key properties—the Poisson property and the independence property— that play an essential role in all settings where the Poisson point process is used.<ref name="ChiuStoyan2013page41and51"/><ref name="Kingman1992page11"/> The two properties are not logically independent; indeed,
===Poisson distribution of point counts===
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:<math> \Pr \{N=n\}=\frac{\Lambda^n}{n!} e^{-\Lambda} </math>
where <math display=inline> n!</math> denotes
By definition, a Poisson point process has the property that the number of points in a bounded region of the process's underlying space is a Poisson-distributed random variable.<ref name="Kingman1992page11">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|pages=11–12}}</ref>
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:<math> \Pr \{N(t)=n\}=\frac{(\lambda t)^n}{n!} e^{-\lambda t}. </math>
The Poisson counting process can also be defined by stating that the time differences between events of the counting process are exponential variables with mean <math display=inline> 1/\lambda</math>.<ref name="Tijms2003"/> The time differences between the events or arrivals are known as '''interarrival''' <ref name="Ross1996page64">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|page=64}}</ref> or '''
===Interpreted as a point process on the real line===
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* the Poisson distribution of the number of arrivals in each interval <math display=inline> (a+t,b+t]</math> only depends on the interval's length <math display=inline> b-a</math>.
In other words, for any finite <math display=inline> t>0</math>, the random variable <math display=inline>
====Law of large numbers====
The quantity <math display=inline> \lambda(b_i-a_i)</math> can be interpreted as the expected or average number of points occurring in the interval <math display=inline> (a_i,b_i]</math>, namely:
:<math> \operatorname E[N(a_i,b_i]
where <math>\operatorname E</math> denotes the [[expected value|expectation]] operator. In other words, the parameter <math display=inline>
:<math> \lim_{t\rightarrow \infty} \frac{N(t)}{t} =\lambda, </math>
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On the real line, the homogeneous Poisson point process has a connection to the theory of [[martingale (probability theory)|martingale]]s via the following characterization: a point process is the homogeneous Poisson point process if and only if
:<math> N(-\infty,t]-\lambda t, </math>
is a martingale.<ref name="merzbach1986characterization">E. Merzbach and D. Nualart. A characterization of the spatial poisson process and changing time. ''The Annals of Probability'', 14(4):1380–1390, 1986.</ref><ref>{{cite journal | url=https://www.jstor.org/stable/3212898 | jstor=3212898 | title=On the Characterization of Point Processes with the Order Statistic Property | last1=Feigin | first1=Paul D. | journal=Journal of Applied Probability | year=1979 | volume=16 | issue=2 | pages=297–304 | doi=10.2307/3212898 | s2cid=123904407 }}</ref>
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{{further|Complete spatial randomness}}
A '''spatial Poisson process''' is a Poisson point process defined in the plane <math>\textstyle \
:<math> \Pr \{N(B)=n\}=\frac{(\lambda|B|)^n}{n!} e^{-\lambda|B|} </math>
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===Defined in higher dimensions===
The previous homogeneous Poisson point process immediately extends to higher dimensions by replacing the notion of area with (high dimensional) volume. For some bounded region <math>\textstyle B</math> of Euclidean space <math>\textstyle \
:<math> \Pr \{N(B)=n\}=\frac{(\lambda|B|)^n}{n!}e^{-\lambda|B|} </math>
where <math>\textstyle |B|</math> now denotes the <math>\textstyle d</math>-dimensional volume of <math>\textstyle B</math>. Furthermore, for a collection of disjoint, bounded Borel sets <math>\textstyle B_1,\dots,B_k \subset \
:<math> \Pr \{N(B_i)=n_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{(\lambda|B_i|)^{n_i}}{n_i!} e^{-\lambda|B_i|}. </math>
Homogeneous Poisson point processes do not depend on the position of the underlying space through its parameter <math>\textstyle \lambda</math>, which implies it is both a stationary process (invariant to translation) and an isotropic (invariant to rotation) stochastic process.<ref name="ChiuStoyan2013page41"/> Similarly to the one-dimensional case, the homogeneous point process is restricted to some bounded subset of <math display=inline>
===Points are uniformly distributed===
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==Inhomogeneous Poisson point process==
[[File:Inhomogeneouspoissonprocess.svg|thumb|Graph of an inhomogeneous Poisson point process on the real line. The events are marked with black crosses, the time-dependent rate <math> \lambda(t) </math> is given by the function marked red.]]
The '''inhomogeneous''' or '''nonhomogeneous''' '''Poisson point process''' (see [[#Terminology|Terminology]]) is a Poisson point process with a Poisson parameter set as some ___location-dependent function in the underlying space on which the Poisson process is defined. For Euclidean space <math>\textstyle \
:<math> \Lambda (B)=\int_B \lambda(x)\,\mathrm dx < \infty, </math>
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===Spatial Poisson process===
An inhomogeneous Poisson process defined in the plane <math>\textstyle \
:<math> \lambda(x,y)= e^{-(x^2+y^2)}, </math>
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===In higher dimensions===
In the plane, <math display="inline"> \Lambda(B)</math> corresponds to
===Applications===
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For the inhomogeneous case, a couple of different methods can be used depending on the nature of the intensity function <math>\textstyle \lambda(x)</math>.<ref name="ChiuStoyan2013page53to55"/> If the intensity function is sufficiently simple, then independent and random non-uniform (Cartesian or other) coordinates of the points can be generated. For example, simulating a Poisson point process on a circular window can be done for an isotropic intensity function (in polar coordinates <math>\textstyle r</math> and <math>\textstyle \theta</math>), implying it is rotationally variant or independent of <math>\textstyle \theta</math> but dependent on <math>\textstyle r</math>, by a change of variable in <math>\textstyle r</math> if the intensity function is sufficiently simple.<ref name="ChiuStoyan2013page53to55"/>
For more complicated intensity functions, one can use an [[Rejection sampling|acceptance-rejection method]], which consists of using (or 'accepting') only certain random points and not using (or 'rejecting') the other points, based on the ratio:.<ref name="Streit2010page14">{{cite book|author=Roy L. Streit|title=Poisson Point Processes: Imaging, Tracking, and Sensing|url=https://books.google.com/books?id=KAWmFYUJ5zsC|date=15 September 2010|publisher=Springer Science & Business Media|isbn=978-1-4419-6923-1|pages=14–16}}</ref>
:<math> \frac{\lambda(x_i)}{\Lambda(W)}=\frac{\lambda(x_i)}{\int_W\lambda(x)\,\mathrm dx. } </math>
where <math>\textstyle x_i</math> is the point under consideration for acceptance or rejection.
That is, a ___location is uniformly randomly selected for consideration, then to determine whether to place a sample at that ___location a uniformly randomly drawn number in <math> [0,1] </math> is compared to the probability density function <math> \frac{\lambda(x)}{\Lambda(W)} </math> , accepting if it is smaller than the probability density function, and repeating until the previously chosen number of samples have been drawn.
==General Poisson point process==
A point process <math>\textstyle {N}</math> is a general Poisson point process with intensity <math>\textstyle \Lambda</math> if it has the two following properties:<ref name="ChiuStoyan2013page51"/>
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===Poisson distribution===
Despite its name, the Poisson point process was neither discovered nor studied by
Poisson derived the Poisson distribution, published in 1841, by examining the binomial distribution in the [[Limit (mathematics)|limit]] of <math>\textstyle p</math> (to zero) and <math>\textstyle n</math> (to infinity). It only appears once in all of Poisson's work,<ref name="stigler1982poisson">{{cite journal |first=S. M. |last=Stigler |title=Poisson on the Poisson Distribution |journal=Statistics & Probability Letters |volume=1 |issue=1 |pages=33–35 |year=1982 |doi=10.1016/0167-7152(82)90010-4 }}</ref> and the result was not well known during his time. Over the following years
<ref name="Stirzaker2000" /> At the end of the 19th century, [[Ladislaus Bortkiewicz]]
===Discovery===
There are a number of claims for early uses or discoveries of the Poisson point process.<ref name="Stirzaker2000"/><ref name="GuttorpThorarinsdottir2012"/> For example, [[John Michell]] in 1767, a decade before Poisson was born, was interested in the probability a star being within a certain region of another star under the erroneous assumption that the stars were "scattered by mere chance", and studied an example consisting of the six brightest [[star]]s in the [[Pleiades]], without deriving the Poisson distribution. This work inspired [[Simon Newcomb]] to study the problem and to calculate the Poisson distribution as an
approximation for the binomial distribution in 1860.<ref name="GuttorpThorarinsdottir2012"/>
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In Sweden 1903, [[Filip Lundberg]] published a thesis containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.<ref name="EmbrechtsFrey2001page367">{{cite book|last1=Embrechts|first1=Paul|title=Stochastic Processes: Theory and Methods|last2=Frey|first2=Rüdiger|last3=Furrer|first3=Hansjörg|chapter=Stochastic processes in insurance and finance|volume=19|year=2001|page=367|issn=0169-7161|doi=10.1016/S0169-7161(01)19014-0|series=Handbook of Statistics|isbn=9780444500144}}</ref><ref name="Cramér1969">{{cite journal|last1=Cramér|first1=Harald|title=Historical review of Filip Lundberg's works on risk theory|journal=Scandinavian Actuarial Journal|volume=1969|issue=sup3|year=1969|pages=6–12|issn=0346-1238|doi=10.1080/03461238.1969.10404602}}</ref>
In [[Denmark]]
In 1910 [[Ernest Rutherford]] and [[Hans Geiger]] published experimental results on counting alpha particles. Their experimental work had mathematical contributions from [[Harry Bateman]], who derived Poisson probabilities as a solution to a family of differential equations, though the solution had been derived earlier, resulting in the independent discovery of the Poisson process.<ref name="Stirzaker2000"/> After this time, there were many studies and applications of the Poisson process, but its early history is complicated, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.<ref name="Stirzaker2000"/>
===Early applications===
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| publisher = [[R (programming language)#CRAN|Comprehensive R Archive Network]]
| doi = 10.18637/jss.v078.i10
| doi-access = free| arxiv = 1612.01907
| s2cid = 14379617
| url = https://cran.r-project.org/web/packages/KFAS/vignettes/KFAS.pdf
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For general point processes, sometimes a subscript on the point symbol, for example <math>\textstyle x</math>, is included so one writes (with set notation) <math>\textstyle x_i\in N</math> instead of <math>\textstyle x\in N</math>, and <math>\textstyle x</math> can be used for the [[bound variable]] in integral expressions such as Campbell's theorem, instead of denoting random points.<ref name="baccelli2009stochastic1"/> Sometimes an uppercase letter denotes the point process, while a lowercase denotes a point from the process, so, for example, the point <math>\textstyle x</math> or <math>\textstyle x_i</math> belongs to or is a point of the point process <math>\textstyle X</math>, and be written with set notation as <math>\textstyle x\in X</math> or <math>\textstyle x_i\in X</math>.<ref name="MollerWaagepetersen2003page7"/>
Furthermore, the set theory and integral or measure theory notation can be used interchangeably. For example, for a point process <math>\textstyle N</math> defined on the Euclidean state space <math>\textstyle {\
:<math> \int_{\
demonstrates two different ways to write a summation over a point process (see also [[Campbell's theorem (probability)]]). More specifically, the integral notation on the left-hand side is interpreting the point process as a random counting measure while the sum on the right-hand side suggests a random set interpretation.<ref name="ChiuStoyan2013page110"/>
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===Laplace functionals===
For a Poisson point process <math>\textstyle N</math> with intensity measure <math>\textstyle \Lambda</math> on some space <math>X</math>, the [[Laplace functional]] is given by:<ref name="baccelli2009stochastic1"/>
:<math> L_N(f)= \mathbb{E} e^{-\
One version of [[Campbell's theorem (probability)#Second definition: Poisson point process|Campbell's theorem]] involves the Laplace functional of the Poisson point process.
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===Probability generating functionals===
The probability generating function of non-negative integer-valued random variable leads to the probability generating functional being defined analogously with respect to any non-negative bounded function <math>\textstyle v</math> on <math>\textstyle \
:<math> G(v)=\operatorname E \left[\prod_{x\in N} v(x) \right] </math>
where the product is performed for all the points in <math display=inline> N </math>. If the intensity measure <math>\textstyle \Lambda</math> of <math>\textstyle {N}</math> is locally finite, then the <math display=inline> G</math> is well-defined for any measurable function <math>\textstyle u</math> on <math>\textstyle \
:<math> G(v)=e^{-\int_{\
which in the homogeneous case reduces to
:<math> G(v)=e^{-\lambda\int_{\
===Moment measure===
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===The Mecke equation===
The Mecke equation characterizes the Poisson point process. Let <math>\mathbb{N}_\sigma</math> be the space of all <math>\sigma</math>-finite measures on some general space <math>\mathcal{Q}</math>. A point process <math>\eta</math> with intensity <math>\lambda</math> on <math>\mathcal{Q}</math> is a Poisson point process if and only if for all measurable functions <math>f:\mathcal{Q}\times\mathbb{N}_\sigma\to \mathbb{R}_+</math> the following holds
:<math>
For further details see.<ref name="Proper Point Process">{{cite book|author1=Günter Last|author2=Mathew Penrose|title=Lectures on the Poisson Process|url=http://www.math.kit.edu/stoch/~last/seite/lectures_on_the_poisson_process/media/lastpenrose2017.pdf|date=8 August 2017}}</ref>
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==Avoidance function==
The '''avoidance function'''
:<math> v(B)=\Pr \{N(B)=0\}. </math>
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Simple point processes are completely characterized by their void probabilities.<ref name="ChiuStoyan2013page43">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|page=43}}</ref> In other words, complete information of a simple point process is captured entirely in its void probabilities, and two simple point processes have the same void probabilities if and if only if they are the same point processes. The case for Poisson process is sometimes known as '''Rényi's theorem''', which is named after [[Alfréd Rényi]] who discovered the result for the case of a homogeneous point process in one-dimension.<ref name="Kingman1992page33">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|pages=34}}</ref>
In one form,<ref name="Kingman1992page33"/> the Rényi's theorem says for a diffuse (or non-atomic) Radon measure <math>\textstyle \Lambda</math> on <math>\textstyle \
}}) that if <math>\textstyle N</math> is a countable subset of <math>\textstyle \
:<math> \Pr \{N(A)=0\} = v(A) = e^{-\Lambda(A)} </math>
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====Displacement theorem====
One version of the displacement theorem<ref name="Kingman1992page61"/> involves a Poisson point process <math>\textstyle {N}</math> on <math>\textstyle \
:<math> \lambda_D(y)=\int_{\
If the Poisson process is homogeneous with <math>\textstyle\lambda(x) = \lambda > 0</math> and if <math>\rho(x, y)</math> is a function of <math>y-x</math>, then
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In other words, after each random and independent displacement of points, the original Poisson point process still exists.
The displacement theorem can be extended such that the Poisson points are randomly displaced from one Euclidean space <math>\textstyle \
===Mapping===
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===Stein's method===
[[Stein's method]] is a mathematical technique originally developed for approximating random variables such as [[Gaussian distribution|Gaussian]] and Poisson variables, which has also been applied to point processes. Stein's method can be used to derive upper bounds on [[probability metric]]s, which give way to quantify how different two random mathematical objects vary stochastically.<ref name="chen2013approximating"/><ref name="barbour1992stein">A. D. Barbour and T. C. Brown. Stein's method and point process approximation. ''Stochastic Processes and their Applications'', 43(1):9–31, 1992.</ref> Upperbounds on probability metrics such as [[total variation]] and [[Wasserstein distance]] have been derived.<ref name="chen2013approximating"/>
Researchers have applied Stein's method to Poisson point processes in a number of ways,<ref name="chen2013approximating"/> such as using [[Palm calculus]].<ref name="chen2004stein"/> Techniques based on Stein's method have been developed to factor into the upper bounds the effects of certain [[point process operation]]s such as thinning and superposition.<ref name="schuhmacher2005super">D. Schuhmacher. Distance estimates for dependent superpositions of point processes. ''Stochastic processes and their applications'', 115(11):1819–1837, 2005.</ref><ref name="schuhmacher2005thinnings">D. Schuhmacher. Distance estimates for poisson process approximations of dependent thinnings. ''Electronic Journal of Probability'', 10:165–201, 2005.</ref> Stein's method has also been used to derive upper bounds on metrics of Poisson and other processes such as the [[Cox point process]], which is a Poisson process with a random intensity measure.<ref name="chen2013approximating"/>
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===Poisson-type random measures===
The [[Poisson-type random measures]] (PT) are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under [[Point process operation#Thinning]]. These random measures are examples of the [[mixed binomial process]] and share the distributional self-similarity property of the [[Poisson random measure]]. They are the only members of the canonical non-negative [[power series]] family of distributions to possess this property and include the [[Poisson distribution]], [[negative binomial distribution]], and [[binomial distribution]]. The Poisson random measure is independent on disjoint subspaces, whereas the other PT random measures (negative binomial and binomial) have positive and negative covariances. The PT random measures are discussed<ref>Caleb Bastian, Gregory Rempala. Throwing stones and collecting bones: Looking for Poisson-like random measures, Mathematical Methods in the Applied Sciences, 2020. [[doi:10.1002/mma.6224]]</ref> and include the [[Poisson random measure]], negative binomial random measure, and binomial random measure.
===Poisson point processes on more general spaces===
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The '''compound Poisson point process''' or '''compound Poisson process''' is formed by adding random values or weights to each point of Poisson point process defined on some underlying space, so the process is constructed from a marked Poisson point process, where the marks form a collection of [[Independent and identically distributed random variables|independent and identically distributed]] non-negative random variables. In other words, for each point of the original Poisson process, there is an independent and identically distributed non-negative random variable, and then the compound Poisson process is formed from the sum of all the random variables corresponding to points of the Poisson process located in some region of the underlying mathematical space.{{sfnp|Daley|Vere-Jones|2003|pages=198–199}}
If there is a marked Poisson point process formed from a Poisson point process <math>\textstyle N</math> (defined on, for example, <math>\textstyle \
:<math> C(B)=\sum_{i=1}^{N(B)} M_i ,</math>
where <math>\textstyle B\subset \
If general random variables <math>\textstyle \{M_i\}</math> take values in, for example, <math>\textstyle d</math>-dimensional Euclidean space <math>\textstyle \
===Failure process with the exponential smoothing of intensity functions===
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