In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, a random measure is a
measure-valued
random element In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansi ...
.
Random measures are for example used in the theory of
random process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es, where they form many important
point processes such as
Poisson point process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
es and
Cox process
In probability theory, a Cox process, also known as a doubly stochastic Poisson process is a point process which is a generalization of a Poisson process where the intensity that varies across the underlying mathematical space (often space or time ...
es.
Definition
Random measures can be defined as
transition kernel In the mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications. Kernels can for example be used to define random measures or stochastic processes. The most important example of kernels ...
s or as
random element In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansi ...
s. Both definitions are equivalent. For the definitions, let
be a
separable complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
and let
be its
Borel -algebra. (The most common example of a separable complete metric space is
)
As a transition kernel
A random measure
is a (
a.s.)
locally finite transition kernel In the mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications. Kernels can for example be used to define random measures or stochastic processes. The most important example of kernels ...
from a (abstract)
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
to
.
Being a transition kernel means that
*For any fixed
, the mapping
:
:is
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
from
to
*For every fixed
, the mapping
:
:is a
measure on
Being locally finite means that the measures
:
satisfy
for all bounded measurable sets
and for all
except some
-
null set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
In the context of
stochastic processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
there is the related concept of a
stochastic kernel, probability kernel, Markov kernel.
As a random element
Define
:
and the subset of locally finite measures by
:
For all bounded measurable
, define the mappings
:
from
to
. Let
be the
-algebra induced by the mappings
on
and
the
-algebra induced by the mappings
on
. Note that
.
A random measure is a random element from
to
that almost surely takes values in
Basic related concepts
Intensity measure
For a random measure
, the measure
satisfying
:
for every positive measurable function
is called the intensity measure of
. The intensity measure exists for every random measure and is a
s-finite measure.
Supporting measure
For a random measure
, the measure
satisfying
:
for all positive measurable functions is called the supporting measure of
. The supporting measure exists for all random measures and can be chosen to be finite.
Laplace transform
For a random measure
, the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
is defined as
: