Random measure

In the theory of measure and probability, a random measure is a random variable whose values ​​are measures . Random geometric structures, as they are examined in stochastic geometry , can be modeled using random dimensions. For example, a point process , such as a general Poisson process , can be viewed as a random measure that assigns the random number of points it contains to a set. In statistics , for example, random measures appear as empirical distributions . Likewise, many point processes such as binomial processes , Poisson processes and Cox processes can be defined as random measures.

definition

Let the -dimensional Euclidean space with Borel σ-algebra and the set of all locally finite measures ( Borel measures ) be on . Further denote the smallest σ-algebra on , so that all mappings , where is a limited Borel set, are measurable . A random measure on is then a random variable on a probability space with values ​​in the measurement space . ${\displaystyle (\mathbb {R} ^{d},{\mathcal {B}}^{d})}$ ${\displaystyle d}$ ${\displaystyle M}$ ${\displaystyle \mu }$ ${\displaystyle (\mathbb {R} ^{d},{\mathcal {B}}^{d})}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle M}$ ${\displaystyle \mu \mapsto \mu (B)}$ ${\displaystyle B}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle X}$ ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle (M,{\mathcal {M}})}$

A random measure thus assigns a measure to every random result, which takes finite values ​​on limited measurable quantities. For any Borel set, is ${\displaystyle \omega \in \Omega }$ ${\displaystyle X_{\omega }}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle A\in {\mathcal {B}}^{d}}$

${\displaystyle X(A)\colon \Omega \to [0,\infty ],\ \omega \mapsto X_{\omega }(A)}$

a nonnegative random variable called the random measure of the set ${\displaystyle A}$ .

Denotes the expected value of , then is by the figure ${\displaystyle \operatorname {E} (X(A))}$ ${\displaystyle X(A)}$

${\displaystyle \operatorname {E} (X)\colon {\mathcal {B}}^{d}\to [0,\infty ],\ A\mapsto \operatorname {E} (X(A))}$

a measure on given the intensity measure of is called. When it is locally finite again, it means integrable . ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle X}$ ${\displaystyle \operatorname {E} (X)}$ ${\displaystyle X}$

example

A random arrangement of points in the plane or in space can be modeled as a random measure: If the positions of points are understood as -value random variables, then by ${\displaystyle Z_{1},Z_{2},\dots ,Z_{N}}$ ${\displaystyle N}$ ${\displaystyle \mathbb {R} ^{d}}$

${\displaystyle X=\delta _{Z_{1}}+\delta _{Z_{2}}+\ldots +\delta _{Z_{N}}}$

a random measure on defined. Here the Dirac measure denotes the place . For a Borel set is then the (random) number of points that lie in the set . ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \delta _{z}}$ ${\displaystyle z}$ ${\displaystyle A\subset \mathbb {R} ^{d}}$ ${\displaystyle X(A)}$ ${\displaystyle A}$

literature

• Olav Kallenberg: Random measures 4th edition, (revised printing of the 3rd edition 1983). Akademie-Verlag et al., Berlin et al. 1986, ISBN 0-12-394960-2 .
• Dietrich Stoyan , Wilfrid S. Kendall, Joseph Mecke: Stochastic Geometry and Its Applications (= Wiley Series in Probability and Statistics ). 2nd Edition. Wiley, Chichester et al. 1995, ISBN 0-471-95099-8 , chap. 7th
• Achim Klenke: Probability Theory. 2nd, corrected edition. Springer, Berlin et al. 2008, ISBN 978-3-540-76317-8 , chap. 24.