Point process

A point process is a special stochastic process and thus an object of investigation of probability theory , a branch of mathematics . Point processes clearly model the random distribution of points, in the simplest case on positive real numbers, in or in more general sets. The best-known example of a point process is the Poisson process , which is also called the Poisson point process . ${\displaystyle \mathbb {R} ^{n}}$

definition

Be a measurable space. A point process is a special case of a random measure . We consider a space , the elements of which are s-finite measures on space . Then is the random variable ${\displaystyle (\mathbb {X} ,{\mathcal {X}})}$ ${\displaystyle \mathrm {M} }$ ${\displaystyle \mathbb {X} }$

${\displaystyle \xi :(\Omega ,{\mathcal {F}})\to (\mathrm {M} ,{\mathcal {M}})}$,

a point process.

Definition on the positive numbers

A sequence of random variables is called a point process (on ) if: ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {R} ^{+}}$

• It is ${\displaystyle X_{0}=0}$
• The sequence is almost certainly strictly increasing monotonically, that is ${\displaystyle X_{0}<X_{1}<X_{2}<\dots }$

example

A simple example of a point process is obtained if one considers an independently identically distributed sequence of random variables that almost certainly take on really positive values. Then you define ${\displaystyle (Y_{k})_{k\in \mathbb {N} }}$

${\displaystyle X_{0}=0}$ and

${\displaystyle X_{n}:=\sum _{i=1}^{n}Y_{i}}$,

so the sequence is increasing monotonically, so it is a point process. ${\displaystyle X_{n}}$

properties

Campbell's formula

The Campbellsche formula describes an important property of a point process to its intensity . The following applies to all -integratable functions ${\displaystyle \xi }$ ${\displaystyle \gamma }$ ${\displaystyle \gamma }$ ${\displaystyle f(x)}$

${\displaystyle \mathbb {E} \left(\int f(x)\xi (\mathrm {d} x)\right)=\int f(x)\gamma (\mathrm {d} x)}$

Real point processes

A distinction is made between real and fake point processes. A point process is then called real when a random variable with values in and random variables exist, so almost certainly applies ${\displaystyle \xi }$ ${\displaystyle M}$ ${\displaystyle \mathbb {N} _{0}\cup \{\infty \}}$ ${\displaystyle (X_{i})_{i\in \mathbb {N} }}$

${\displaystyle \xi (B)=\sum \limits _{n=1}^{M}\delta _{X_{i}}(B)\quad \forall B\in {\mathcal {X}}}$

It can be shown that for every Poisson point process there is a real point process which has the same distribution in the same space.

Explanation

A point process on models the random distribution of points on the positive numbers. The first part of the definition says that the first point should be the zero point. The second part says that the points are provided with an order, so they are already sorted according to size. ${\displaystyle \mathbb {R} ^{+}}$

In the example above, the random variables are defined via their increases. The distributions of the increases correspond, in this example , in the general case , to the distribution of the distance between the points. For example, in the Poisson process, the distances between two points are exponentially distributed . ${\displaystyle X_{n}}$ ${\displaystyle X_{n+1}-X_{n}=Y_{n}}$ ${\displaystyle X_{n+1}-X_{n}}$

The associated counting process

Every point process on lets itself through ${\displaystyle \mathbb {R} ^{+}}$

${\displaystyle N_{t}=\sum _{i=1}^{\infty }\mathbf {1} _{\{X_{i}\leq t\}}}$

assign a counting process ( here denotes the characteristic function on the set ). The counting process clearly runs from zero to the positive numbers at a constant speed and counts how many points it has already hit up to that point in time . The counting process and the dot process highlight two aspects of the same idea here. In their formalization, however, they differ significantly, as can already be seen from their index set. ${\displaystyle \mathbf {1} _{A}}$ ${\displaystyle A}$ ${\displaystyle t}$

Web links

• Yu.K. Belyaev: Stochastic point process . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Stover, Christopher: Point Process . In: MathWorld (English).

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 358 , doi : 10.1007 / 978-3-642-41997-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 278-279 , doi : 10.1007 / b137972 .