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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. The point process has convenient mathematical properties, 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, biology, ecology, geology, physics, economics, image processing, and telecommunications.
Contents
- Overview of definitions
- Poisson distribution
- Complete independence
- Different definitions
- Homogeneous Poisson point process
- Interpreted as a counting process
- Interpreted as a point process on the real line
- Key properties
- Law of large numbers
- Memoryless property
- Orderliness and simplicity
- Martingale characterization
- Relationship to other processes
- Restricted to the half line
- Applications
- Generalizations
- Spatial Poisson point process
- Defined in higher dimensions
- Points are uniformly distributed
- Inhomogeneous Poisson point process
- Defined on the real line
- Counting process interpretation
- Spatial Poisson process
- In higher dimensions
- Interpretation of the intensity function
- Simple point process
- Simulation
- Step 1 Number of points
- Homogeneous case
- Inhomogeneous case
- Step 2 Positioning of points
- General Poisson point process
- Discovery
- Early applications
- History of terms
- Terminology
- Notation
- Functionals and moment measures
- Laplace functionals
- Probability generating functionals
- Moment measure
- Factorial moment measure
- Avoidance function
- Rnyis theorem
- Point process operations
- Thinning
- Superposition
- Superposition theorem
- Clustering
- Random displacement
- Displacement theorem
- Mapping
- Mapping theorem
- Approximations with Poisson point processes
- Clumping heuristic
- Steins method
- Convergence to a Poisson point process
- Generalizations of Poisson point processes
- Poisson point processes on more general spaces
- Cox point process
- Marked Poisson point process
- Marking theorem
- Compound Poisson point process
- References
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 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, can represent the locations of scattered objects such as transmitters in a wireless network, 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, stochastic geometry, spatial statistics and continuum percolation theory. On 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 for 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.
The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. Its name derives from 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 a random variable with a Poisson distribution. The process was discovered independently and repeatedly in different settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics.
The point process depends on a single mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings, a Radon measure. 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. 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. 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.
Overview of definitions
The Poisson point process is one of the most studied and used 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 as well definitions of varying generality owing to its many applications and characterizations.
A Poisson point process is defined on some underlying mathematical space, called a carrier space, or state space, though the latter term has a different meaning in the context of stochastic processes. 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; in higher dimensions such as the plane where it plays a role in stochastic geometry and spatial statistics; or on more general mathematical spaces. 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.
Poisson distribution
The Poisson point process is related to the Poisson distribution, which implies that the probability of a Poisson random variable
where
Complete independence
For a collection of disjoint and bounded subregions of the underlying space, the number of points of a Poisson point process in each bounded subregion will be completely independent of all the others. This property is known under several names such as complete randomness, complete independence, or independent scattering and is common to all Poisson point processes. In other words, there is a lack of interaction between different regions and the points in general, which motivates the Poisson process being sometimes called a purely or completely random process.
Different definitions
For all the instances of the Poisson point process, the two key properties of the Poisson distribution and complete independence play an important role.
Homogeneous Poisson point process
If a Poisson point process has a constant parameter, say,
Interpreted as a counting process
The homogeneous Poisson point process, when considered on the positive half-line, can be defined as a counting process, a type of stochastic process, which can be denoted as
The last property implies:
In other words, the probability of the random variable
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
Interpreted as a point process on the real line
Interpreted as a point process, a Poisson point process can be defined on the real line by first considering number of points of a point process being in the half-open interval points of the process in the interval
For some positive integer
where the real numbers
In other words,
Key properties
The previous definition has two important features shared by Poisson point processes in general:
Furthermore, it has a third feature related to just the homogeneous Poisson point process:
In other words, for any finite
Law of large numbers
The quantity
where
where
Memoryless property
The distance between two consecutive points of a point process on the real line will be an exponential random variable with parameter
Orderliness and simplicity
A point process with stationary increments is sometimes said to be orderly, ordinary, or regular if:
where little-o notation is being used. A point process is called a simple point process when the probability of any of its two points coinciding in the same position, on the underlying space, is zero. For point processes in general on the real line, the property of orderliness implies that the process is simple, which is the case for the homogeneous Poisson point process.
Martingale characterization
On the real line, the homogeneous Poisson point process has a connection to the theory of martingales via the following characterization: a point process is the homogeneous Poisson point process if and only if
is a martingale.
Relationship to other processes
On the real line, the Poisson process is a type of continuous-time Markov process known as a birth-death process (with just births and zero deaths) and is called a pure or simple birth process. More complicated processes with the Markov property, such as Markov arrival processes, have been defined where the Poisson process is a special case.
Restricted to the half-line
If the homogeneous Poisson process is considered just on the half-line
Applications
There have been many applications of the homogeneous Poisson process on the real line in an attempt to model seemingly random and independent events occurring. It has a fundamental role in queueing theory, which is the probability field of developing suitable stochastic models to represent the random arrival and departure of certain phenomena. For example, customers arriving and being served or phone calls arriving at a phone exchange can be both studied with techniques from queueing theory. The original paper proposing the online payment system known as Bitcoin featured a mathematical model based on a homogeneous Poisson point process.
Generalizations
The homogeneous Poisson process on the real line is considered one of the simplest stochastic processes for counting random numbers of points. This process can be generalized in a number of ways. One possible generalization is to extend the distribution of interarrival times from the exponential distribution to other distributions, which introduces the stochastic process known as a renewal process. Another generalization is to define the Poisson point process on higher dimensional spaces such as the plane.
Spatial Poisson point process
A spatial Poisson process is a Poisson point process defined in the plane
where
For some finite integer
Applications
The spatial Poisson point process features prominently in spatial statistics, stochastic geometry, and continuum percolation theory. This point process is applied in various physical sciences such as a model developed for alpha particles being detected. In recent years, it has been frequently used to model seemingly disordered spatial configurations of certain wireless communication networks. For example, models for cellular or mobile phone networks have been developed where it is assumed the phone network transmitters, known as base stations, are positioned according to a homogeneous Poisson point process.
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
where
Homogeneous Poisson point processes do not depend on the position of the underlying space through its parameter
Points are uniformly distributed
If the homogeneous point process is defined on the real line as a mathematical model for occurrences of some phenomenon, then it has the characteristic that the positions of these occurrences or events on the real line (often interpreted as time) will be uniformly distributed. More specifically, if an event occurs (according to this process) in an interval
Inhomogeneous Poisson point process
The inhomogeneous or nonhomogeneous Poisson point process (see 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
where
Furthermore,
Defined on the real line
On the real line, the inhomogeneous or non-homogeneous Poisson point process has mean measure given by a one-dimensional integral. For two real numbers
where the mean or intensity measure is:
which means that the random variable
A feature of the one-dimension setting, is that an inhomogeneous Poisson process can be transformed into a homogeneous by a monotone transformation or mapping, which is achieved with the inverse of
Counting process interpretation
The inhomogeneous Poisson point process, when considered on the positive half-line, is also sometimes defined as a counting process. With this interpretation, the process, which is sometimes written as
where
The above properties imply that
which implies
Spatial Poisson process
An inhomogeneous Poisson process defined in the plane
so the corresponding intensity measure is given by the surface integral
where
In higher dimensions
In the plane,
Applications
When the real line is interpreted as time, the inhomogeneous process is used in the fields of counting processes and in queueing theory. Examples of phenomena which have been represented by or appear as an inhomogeneous Poisson point process include:
In the plane, the Poisson point process is important in the related disciplines of stochastic geometry and spatial statistics. The intensity measure of this point process is dependent on the location of underlying space, which means it can be used to model phenomena with a density that varies over some region. In other words, the phenomena can be represented as points that have a location-dependent density. This processes has been used in various various disciplines and uses include the study of salmon and sea lice in the oceans, forestry, and search problems.
Interpretation of the intensity function
The Poisson intensity function
For example, given a homogeneous Poisson point process on the real line, the probability of finding a single point of the process in a small interval of width
Simple point process
If a Poisson point process has an intensity measure that is a locally finite and diffuse (or non-atomic), then it is a simple point process. For a simple point process, the probability of a point existing at a single point or location in the underlying (state) space is either zero or one. This implies that, with probability one, no two (or more) points of a Poisson point process coincide in location in the underlying space.
Simulation
Simulating a Poisson point process on a computer is usually done in a bounded region of space, known as a simulation window, and requires two steps: appropriately creating a random number of points and then suitably placing the points in a random manner. Both these two steps depend on the specific Poisson point process that is being simulated.
Step 1: Number of points
The number of points
Homogeneous case
For the homogeneous case with the constant
Inhomogeneous case
For the inhomogeneous case,
Step 2: Positioning of points
The second stage requires randomly placing the
Homogeneous case
For the homogeneous case in one dimension, all points are uniformly and independently placed in the window or interval
Inhomogeneous case
For the inhomogeneous, a couple of different methods can be used depending on the nature of the intensity function
For more complicated intensity functions, one can use an 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:
where
General Poisson point process
The Poisson point process can be further generalized to what is sometimes known as the general Poisson point process or general Poisson process by using a Radon measure
A point process
The Radon measure
Furthermore, if
where the density
Poisson distribution
Despite its name, the Poisson point process was neither discovered nor studied by the French mathematician Siméon Denis Poisson; the name is cited as an example of Stigler's law. The name stems from its inherent relation to the Poisson distribution, derived by Poisson as a limiting case of the binomial distribution. This describes the probability of the sum of
Poisson derived the Poisson distribution, published in 1841, by examining the binomial distribution in the limit of
Discovery
There are a number of claims for early uses or discoveries of the Poisson point process. 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 assumption that the stars were "scattered by mere chance", and studied an example consisting of the six brightest stars 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.
At the beginning of the 20th century the Poisson process (in one dimension) would arise independently in different situations. 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.
In Denmark in 1909 another discovery occurred when A.K. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.
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. 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.
Early applications
The years after 1909 led to a number of studies and applications of the Poisson point process, however, its early history is complex, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and others working in the physical sciences. The early results were published in different languages and in different settings, with no standard terminology and notation used. For example, in 1922 Swedish chemist and Nobel Laureate Theodor Svedberg proposed a model in which a spatial Poisson point process is the underlying process in order to study how plants are distributed in plant communities. A number of mathematicians started studying the process in the early 1930s, and important contributions were made by Andrey Kolmogorov, William Feller and Aleksandr Khinchin, among others. As an application, Kolmogorov used a spatial Poisson point process to model the formation of crystals in metals. In the field of teletraffic engineering, mathematicians and statisticians studied and used Poisson and other point processes.
History of terms
The Swede Conny Palm in his 1943 dissertation studied the Poisson and other point processes in the one-dimensional setting by examining them in terms of the statistical or stochastic dependence between the points in time. In his work exists the first known recorded use of the term point process as Punktprozesse in German.
It is believed that William Feller was the first in print to refer to it as the Poisson process in a 1940 paper. Although the Swede Ove Lundberg used the term Poisson process in his 1940 PhD dissertation, in which Feller was acknowledged as an influence, it has been claimed that Feller coined the term before 1940. It has been remarked that both Feller and Lundberg used the term as though it were well-known, implying it was already in spoken use by then. Feller worked from 1936 to 1939 alongside Harald Cramér at Stockholm University, where Lundberg was a PhD student under Cramér who did not use the term Poisson process in a book by him, finished in 1936, but did in subsequent editions, which his has led to the speculation that the term Poisson process was coined sometime between 1936 and 1939 at the Stockholm University.
Terminology
The terminology of point process theory in general has been criticized for being too varied. In addition to the word point often being omitted, the homogeneous Poisson (point) process is also called a stationary Poisson (point) process, as well as uniform Poisson (point) process. The inhomogeneous Poisson point process, as well as being called nonhomogeneous, is also referred to as the non-stationary Poisson process.
The term point process has been criticized, as the term proces can suggest over over time and space, so random point field, resulting in the terms Poisson random point field or Poisson point field being also used. A point process is considered, and sometimes called, a random counting measure, hence the Poisson point process is also referred to as a Poisson random measure, a term used in the study of Lévy processes, but some choose to use the two terms for Poisson points processes defined on two different underlying spaces.
The measure
The extent of the Poisson point process is sometimes called the exposure.
Notation
The notation of the Poisson point process depends on its setting and the field it is being applied in. For example, on the real line, the Poisson process, both homogeneous or inhomogeneous, is sometimes interpreted as a counting process, and the notation
Another reason for varying notation is due to the theory of point processes, which has a couple of mathematical interpretations. For example, a simple Poisson point process may be considered as a random set, which suggests the notation
For general point processes, sometimes a subscript on the point symbol, for example
Furthermore, the set theory and integral or measure theory notation can be used interchangeably. For example, for a point process
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.
Functionals and moment measures
In probability theory, operations are applied to random variables for different purposes. Sometimes these operations are regular expectations that produce the average or variance of a random variable. Others, such as characteristic functions (or Laplace transforms) of a random variable can be used to uniquely identify or characterize random variables and prove results like the central limit theorem. In the theory of point processes there exist analogous mathematical tools which usually exist in the forms of measures and functionals instead of moments and functions respectively.
Laplace functionals
For a Poisson point process
One version of Campbell's theorem involves the Laplace functional of the Poisson point process.
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
where the product is performed for all the points in
which in the homogeneous case reduces to
Moment measure
For a general Poisson point process with intensity measure
which for a homogeneous Poisson point process with constant intensity
where
Factorial moment measure
For a general Poisson point process with intensity measure
where
For a homogeneous Poisson point process the
where
Avoidance function
The avoidance function or void probability
For a general Poisson point process
Rényi's theorem
Simple point processes are completely characterized by their void probabilities. 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 voiud 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.
In one form, the Rényi's theorem says for a diffuse (or non-atomic) Radon measure
then
Point process operations
Mathematical operations can be performed on point processes in order to new point processes and develop new mathematical models for the locations of certain objects. One example of an operation is known as thinning which entails deleting or removing the points of some point process according to a rule, creating a new process with the remaining points (the deleted points also form a point process).
Thinning
For the Poisson process, the independent
Furthermore, after randomly thinning a Poisson point process, the kept or remaining points also form a Poisson point process, which has the intensity measure
The two separate Poisson point processes formed respectively from the removed and kept points are stochastically independent of each other. In other words, if a region is known to contain
Superposition
If there is a countable collection of point processes
also forms a point process. In other words, any points located in any of the point processes
Superposition theorem
The Superposition theorem of the Poisson point process says that the superposition of independent Poisson point processes
In other words, the union of two (or countably more) Poisson processes is another Poisson process. If a point
For two homogeneous Poisson processes with intensities
and
Clustering
The operation clustering is performed when each point
Random displacement
A mathematical model may require randomly moving points of a point process to other locations on the underlying mathematical space, which gives rise to a point process operation known as displacement or translation. The Poisson point process has been used to model, for example, the movement of plants between generations, owing to the displacement theorem, which loosely says that the random independent displacement of points of a Poisson point process (on the same underlying space) forms another Poisson point process.
Displacement theorem
One version of the displacement theorem involves a Poisson point process
which for the homogeneous case with a constant
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
Mapping
Another property that is considered useful is the ability to map a Poisson point process from one underlying space to another space.
Mapping theorem
If the mapping (or transformation) adheres to some conditions, then the resulting mapped (or transformed) collection of points also form a Poisson point process, and this result is sometimes referred to as the Mapping theorem. The theorem involves some Poisson point process with mean measure
More specifically, one can consider a (Borel measurable) function
with no atoms, where
Approximations with Poisson point processes
The tractability of the Poisson process means that sometimes it is convenient to approximate a non-Poisson point process with a Poisson one. The overall aim is to approximate the both number of points of some point process and the location of each point by a Poisson point process. There a number of methods that can be used to justify, informally or rigorously, approximating the occurrence of random events or phenomena with suitable Poisson point processes. The more rigorous methods involve deriving upper bounds on the probability metrics between the Poisson and non-Poisson point processes, while other methods can be justified by less formal heuristics.
Clumping heuristic
One method for approximating random events or phenomena with Poisson processes is called the clumping heuristic. The general heuristic or principle involves using the Poisson point process (or Poisson distribution) to approximate events, which are considered rare or unlikely, of some stochastic process. In some cases these rare events are close to independent, hence a Poisson point process can be used. When the events are not independent, but tend to occur in clusters or clumps, then if these clumps are suitably defined such that they are approximately independent of each other, then the number of clumps occurring will be close to a Poisson random variable and the locations of the clumps will be close to a Poisson process.
Stein's method
Stein's method, a rigorous mathematical technique originally developed for approximating random variables such as Gaussian and Poisson variables, has also been developed and applied to point processes. Stein's method can be used to derive upper bounds on probability metrics, which give way to quantify how different two random mathematical objects vary stochastically, of the Poisson and other point processes. Upperbounds on probability metrics such as total variation and Wasserstein distance have been derived.
Researchers have applied Stein's method to Poisson point processes in a number of ways, such as using Palm calculus. Techniques based on Stein's method have been developed to factor into the upper bounds the effects of certain point process operations such as thinning and superposition. 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.
Convergence to a Poisson point process
In general, when an operation is applied to a general point process the resulting process is usually not a Poisson point process. For example, if a point process, other than a Poisson, has its points randomly and independently displaced, then the process would not necessarily be a Poisson point process. However, under certain mathematical conditions for both the original point process and the random displacement, it has been shown via limit theorems that if the points of a point process are repeatedly displaced in a random and independent manner, then the finite-distribution of the point process will converge (weakly) to that of a Poisson point process.
Similar convergence results have been developed for thinning and superposition operations that show that such repeated operations on point processes can, under certain conditions, result in the process converging to a Poisson point processes, provided a suitable rescaling of the intensity measure (otherwise values of the intensity measure of the resulting point processes would approach zero or infinity). Such convergence work is directly related to the results known as the Palm–Khinchin equations, which has its origins in the work of Conny Palm and Aleksandr Khinchin, and help explains why the Poisson process can often be used as a mathematical model of various random phenomena.
Generalizations of Poisson point processes
The Poisson point process can be generalized by, for example, changing its intensity measure or defining on more general mathematical spaces. These generalizations can be studied mathematically as well as used to mathematically model or represent physical phenomena.
Poisson point processes on more general spaces
For mathematical models the Poisson point process is often defined in Euclidean space, but has been generalized to more abstract spaces and plays a fundamental role in the study of random measures, which requires an understanding of mathematical fields such as probability theory, measure theory and topology.
In general, the concept of distance is of practical interest for applications, while topological structure is needed for Palm distributions, meaning that point processes are usually defined on mathematical spaces with metrics. Furthermore, a realization of a point process can be considered as a counting measure, so points processes are types of random measures known as random counting measures. In this context, the Poisson and other point processes has been studied on a locally compact second countable Hausdorff space.
Cox point process
A Cox process point, Cox process or doubly stochastic Poisson process is a generalization of the Poisson point process by letting its intensity measure
Marked Poisson point process
For a given point process, each random point of a point process can have a random mathematical object, known as a mark, randomly assigned to it. These marks can be as diverse as integers, real numbers, lines, geometrical objects or other point processes. The pair consisting of a point of the point process and its corresponding mark is called a marked point, and all the marked points form a marked point process. It is often assumed that the random marks are independent of each other and identically distributed, yet the mark of a point can still depend on the location of its corresponding point in the underlying (state) space. If the underlying point process is a Poisson point process, then the resulting point process is a marked Poisson point process.
Marking theorem
If a general point process is defined on some mathematical space and the random marks are defined on another mathematical space, then the marked point process is defined on the Cartesian product of these two spaces. For a marked Poisson point process with independent and identically distributed marks, the Marking theorem states that this marked point process is also a (non-marked) Poisson point process defined on the aforementioned Cartesian product of the two mathematical spaces, which is not true for general point processes.
Compound Poisson point process
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 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 then formed from the sum of all the random variables corresponding to points of the Poisson process located in a some region of the underlying mathematical space.
If there is a marked Poisson point processes formed from a Poisson point process
where
If general random variables