Intensity measure

In mathematics, an intensity measure is a measure that is assigned to a random measure . The intensity measure corresponds to the expected value of the random measure and thus indicates which volume the random measure assigns to a certain amount on average. The intensity measure thus contains important information about the random measure. For example, Poisson processes are already clearly defined by specifying their intensity measure.

definition

Given a random dimension on the measuring room . This means that it almost certainly assumes finite measures locally as values. ${\displaystyle X}$ ${\displaystyle (S,{\mathcal {A}})}$ ${\displaystyle X}$ ${\displaystyle (S,{\mathcal {A}})}$

Then the measure is called on , that through ${\displaystyle \operatorname {EX} }$ ${\displaystyle (S,{\mathcal {A}})}$

${\displaystyle \operatorname {EX} \colon {\mathcal {A}}\to [0,+\infty ]}$,

${\displaystyle \operatorname {EX} \colon A\mapsto \operatorname {E} [X(A)]}$

is given, the intensity measure of . A distinction must be made here between the designation of the intensity measure as and the formation of the expected value of a random variable by . ${\displaystyle X}$ ${\displaystyle \operatorname {EX} }$ ${\displaystyle Y}$ ${\displaystyle \operatorname {E} [Y]}$

Examples

In a binomial process given by and a distribution is considered by construction . With the elementary properties of the binomial distribution then follows directly ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle \nu }$ ${\displaystyle X(A)\sim \operatorname {Bin} _{n,\nu (A)}}$

${\displaystyle \operatorname {EX} (A)=\operatorname {E} [X(A)]=n\nu (A)}$.

So the intensity measure of a binomial process is given by

${\displaystyle \operatorname {EX} =n\nu }$.

properties

The intensity measure is always s-finite and fulfilled ${\displaystyle \operatorname {EX} }$

${\displaystyle \operatorname {E} \left[\int f(x)X(\mathrm {d} x)\right]=\int f(x)\operatorname {EX} (dx)}$

for every positive measurable function . ${\displaystyle (S,{\mathcal {A}})}$

literature

• Olav Kallenberg: Random Measures, Theory and Applications . Springer, Switzerland 2017, doi : 10.1007 / 978-3-319-41598-7 .
• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .

Individual evidence

1. ↑ Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
2. ^ Olav Kallenberg: Random Measures, Theory and Applications . Springer, Switzerland 2017, p. 53 , doi : 10.1007 / 978-3-319-41598-7 .