Random measure

Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent. For the definitions, let $E$ be a separable complete metric space and let $\backslash mathcal\; E$ be its Borel $\backslash sigma$-algebra. (The most common example of a separable complete metric space is $\backslash R^n$.)

A random measure $\backslash zeta$ is a (a.s.) locally finite transition kernel from an abstract probability space $(\backslash Omega,\; \backslash mathcal\; A,\; P)$ to $(E,\; \backslash mathcal\; E)$.

Being a transition kernel means that

• For any fixed $B\; \backslash in\; \backslash mathcal\; \backslash mathcal\; E$, the mapping

:$\backslash omega\; \backslash mapsto\; \backslash zeta(\backslash omega,B)$

:is measurable from $(\backslash Omega,\; \backslash mathcal\; A)$ to $(\backslash R,\; \backslash mathcal\; B(\backslash R))$

• For every fixed $\backslash omega\; \backslash in\; \backslash Omega$, the mapping

:$B\; \backslash mapsto\; \backslash zeta(\backslash omega,\; B)\; \backslash quad\; (B\; \backslash in\; \backslash mathcal\; E)$

:is a measure on $(E,\; \backslash mathcal\; E)$

Being locally finite means that the measures

:$B\; \backslash mapsto\; \backslash zeta(\backslash omega,\; B)$

satisfy for all bounded measurable sets $\backslash tilde\; B\; \backslash in\; \backslash mathcal\; E$

and for all $\backslash omega\; \backslash in\; \backslash Omega$ except some $P$-null set

In the context of stochastic processes there is the related concept of a stochastic kernel, probability kernel, Markov kernel.

Define

:$\backslash tilde\; \backslash mathcal\; M:=\; \backslash \{\; \backslash mu\; \backslash mid\; \backslash mu\; \backslash text\{\; is\; measure\; on\; \}\; (E,\; \backslash mathcal\; E)\; \backslash \}$

and the subset of locally finite measures by

:

For all bounded measurable $\backslash tilde\; B$, define the mappings

:$I\_\{\backslash tilde\; B\; \}\; \backslash colon\; \backslash mu\; \backslash mapsto\; \backslash mu(\backslash tilde\; B)$

from $\backslash tilde\; \backslash mathcal\; M$ to $\backslash R$. Let $\backslash tilde\; \backslash mathbb\; M$ be the $\backslash sigma$-algebra induced by the mappings $I\_\{\backslash tilde\; B\; \}$ on $\backslash tilde\; \backslash mathcal\; M$ and $\backslash mathbb\; M$ the $\backslash sigma$-algebra induced by the mappings $I\_\{\backslash tilde\; B\; \}$ on $\backslash mathcal\; M$. Note that $\backslash tilde\backslash mathbb\; M|\_\{\backslash mathcal\; M\}=\; \backslash mathbb\; M$.

A random measure is a random element from $(\backslash Omega,\; \backslash mathcal\; A,\; P)$ to $(\backslash tilde\; \backslash mathcal\; M,\; \backslash tilde\; \backslash mathbb\; M)$ that almost surely takes values in $(\backslash mathcal\; M,\; \backslash mathbb\; M)$

{{main article|intensity measure}}

For a random measure $\backslash zeta$, the measure $\backslash operatorname\; E\; \backslash zeta$ satisfying

:$\backslash operatorname\; E\; \backslash left[\; \backslash int\; f(x)\; \backslash ;\; \backslash zeta\; (\backslash mathrm\; dx\; )\backslash right]\; =\; \backslash int\; f(x)\; \backslash ;\; \backslash operatorname\; E\; \backslash zeta\; (\backslash mathrm\; dx)$

for every positive measurable function $f$ is called the intensity measure of $\backslash zeta$. The intensity measure exists for every random measure and is a s-finite measure.

For a random measure $\backslash zeta$, the measure $\backslash nu$ satisfying

:$\backslash int\; f(x)\; \backslash ;\; \backslash zeta(\backslash mathrm\; dx\; )=0\; \backslash text\{\; a.s.\; \}\; \backslash text\{\; iff\; \}\; \backslash int\; f(x)\; \backslash ;\; \backslash nu\; (\backslash mathrm\; dx)=0$

for all positive measurable functions is called the supporting measure of $\backslash zeta$. The supporting measure exists for all random measures and can be chosen to be finite.

For a random measure $\backslash zeta$, the Laplace transform is defined as

:$\backslash mathcal\; L\_\backslash zeta(f)=\; \backslash operatorname\; E\; \backslash left[\; \backslash exp\; \backslash left(\; -\backslash int\; f(x)\; \backslash ;\; \backslash zeta\; (\backslash mathrm\; dx\; )\; \backslash right)\; \backslash right]$

for every positive measurable function $f$.

For a random measure $\backslash zeta$, the integrals

:$\backslash int\; f(x)\; \backslash zeta(\backslash mathrm\; dx)$

and

$\backslash zeta(A)\; :=\; \backslash int\; \backslash mathbf\; 1\_A(x)\; \backslash zeta(\backslash mathrm\; dx)$

for positive $\backslash mathcal\; E$-measurable $f$ are measurable, so they are random variables.

The distribution of a random measure is uniquely determined by the distributions of

:$\backslash int\; f(x)\; \backslash zeta(\backslash mathrm\; dx)$

for all continuous functions with compact support $f$ on $E$. For a fixed semiring $\backslash mathcal\; I\; \backslash subset\; \backslash mathcal\; E$ that generates $\backslash mathcal\; E$ in the sense that $\backslash sigma(\backslash mathcal\; I)=\backslash mathcal\; E$, the distribution of a random measure is also uniquely determined by the integral over all positive simple $\backslash mathcal\; I$-measurable functions $f$.

A measure generally might be decomposed as:

:$\backslash mu=\backslash mu\_d\; +\; \backslash mu\_a\; =\; \backslash mu\_d\; +\; \backslash sum\_\{n=1\}^N\; \backslash kappa\_n\; \backslash delta\_\{X\_n\},$

Here $\backslash mu\_d$ is a diffuse measure without atoms, while $\backslash mu\_a$ is a purely atomic measure.

A random measure of the form:

:$\backslash mu=\backslash sum\_\{n=1\}^N\; \backslash delta\_\{X\_n\},$

where $\backslash delta$ is the Dirac measure and $X\_n$ are random variables, is called a point process or random counting measure. This random measure describes the set of N particles, whose locations are given by the (generally vector valued) random variables $X\_n$. The diffuse component $\backslash mu\_d$ is null for a counting measure.

In the formal notation of above a random counting measure is a map from a probability space to the measurable space {{nowrap|($N\_X$, $\backslash mathfrak\{B\}(N\_X)$)}}. Here $N\_X$ is the space of all boundedly finite integer-valued measures $N\; \backslash in\; M\_X$ (called counting measures).

The definitions of expectation measure, Laplace functional, moment measures and stationarity for random measures follow those of point processes. Random measures are useful in the description and analysis of Monte Carlo methods, such as Monte Carlo numerical quadrature and particle filters.

• Poisson random measure
• Vector measure
• Ensemble

{{cite book |last1=Klenke |first1=Achim |year=2008 |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6 |page=526}}

{{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|series=Probability Theory and Stochastic Modelling |volume=77 |location= Switzerland |publisher=Springer |page=1|doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3}}

"Crisan, D., Particle Filters: A Theoretical Perspective, in Sequential Monte Carlo in Practice, Doucet, A., de Freitas, N. and Gordon, N. (Eds), Springer, 2001, {{ISBN|0-387-95146-6}}

{{Cite book | doi = 10.1007/b97277 | first1 = D. J. | last1 = Daley | first2 = D. | last2 = Vere-Jones| title = An Introduction to the Theory of Point Processes | series = Probability and its Applications | year = 2003 | isbn = 0-387-95541-0 }}

Kallenberg, O., Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986). {{ISBN|0-12-394960-2}} {{MR|854102}}. An authoritative but rather difficult reference.

Jan Grandell, Point processes and random measures, Advances in Applied Probability 9 (1977) 502-526. {{MR|0478331}} [https://www.jstor.org/stable/1426111 JSTOR] A nice and clear introduction.

{{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|series=Probability Theory and Stochastic Modelling |volume=77 |location= Switzerland |publisher=Springer |page=52|doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3}}

{{Measure theory}}

Category:Measures (measure theory)

Category:Stochastic processes