Poisson-type random measures are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under thinning. They are the only distributions in the canonical non-negative power series family of distributions to possess this property and include the
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
,
negative binomial distribution, and
binomial distribution. The PT family of distributions is also known as the Katz family of distributions, the Panjer or
(a,b,0) class of distributions
In probability theory, a member of the (''a'', ''b'', 0) class of distributions is any distribution of a discrete random variable ''N'' whose values are nonnegative integers whose probability mass function satisfies the recurrence formula
: \fra ...
and may be retrieved through the
Conway–Maxwell–Poisson distribution
In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM–Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the P ...
.
Throwing stones
Let
be a non-negative integer-valued random variable
) with law
, mean
and when it exists variance
. Let
be a probability measure on the
measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then the ...
. Let
be a collection of iid random variables (stones) taking values in
with law
.
The random counting measure
on
depends on the pair of deterministic probability measures
through the stone throwing construction (STC)
where
has law
and iid
have law
.
is a
mixed binomial process
Let
be the collection of positive
-measurable functions. The probability law of
is encoded in the
Laplace functional
where
is the generating function of
. The mean and variance are given by
and
The covariance for arbitrary
is given by
When
is Poisson, negative binomial, or binomial, it is said to be Poisson-type (PT). The joint distribution of the collection
is for
and
:
The following result extends construction of a random measure
to the case when the collection
is expanded to
where
is a random transformation of
. Heuristically,
represents some properties (marks) of
. We assume that the conditional law of
follows some transition kernel according to
.
Theorem: Marked STC
Consider random measure
and the transition probability kernel
from
into
. Assume that given the collection
the variables
are conditionally independent with
. Then
is a random measure on
. Here
is understood as
. Moreover, for any
we have that
where
is pgf of
and
is defined as
The following corollary is an immediate consequence.
Corollary: Restricted STC
The quantity
is a well-defined random measure on the measurable subspace
where
and
. Moreover, for any
, we have that
where
.
Note
where we use
.
Collecting Bones
The probability law of the random measure is determined by its Laplace functional and hence generating function.
Definition: Bone
Let
be the counting variable of
restricted to
. When
and
share the same family of laws subject to a rescaling
of the parameter
, then
is a called a bone distribution. The bone condition for the pgf is given by
.
Equipped with the notion of a bone distribution and condition, the main result for the existence and uniqueness of Poisson-type (PT) random counting measures is given as follows.
Theorem: existence and uniqueness of PT random measures
Assume that
with pgf
belongs to the canonical non-negative power series (NNPS) family of distributions and
. Consider the random measure
on the space
and assume that
is diffuse. Then for any
with
there exists a mapping
such that the restricted random measure is
, that is,
iff
is Poisson, negative binomial, or binomial (Poisson-type).
The proof for this theorem is based on a generalized additive Cauchy equation and its solutions. The theorem states that out of all NNPS distributions, only PT have the property that their restrictions
share the same family of distribution as
, that is, they are closed under thinning. The PT random measures are the
Poisson random measure Let (E, \mathcal A, \mu) be some measure space with \sigma-finite measure \mu. The Poisson random measure with intensity measure \mu is a family of random variables \_ defined on some probability space (\Omega, \mathcal F, \mathrm) such that
i) ...
, negative binomial random measure, and binomial random measure. Poisson is additive with independence on disjoint sets, whereas negative binomial has positive covariance and binomial has negative covariance. The
binomial process A binomial process is a special point process in probability theory.
Definition
Let P be a probability distribution and n be a fixed natural number. Let X_1, X_2, \dots, X_n be i.i.d. random variables with distribution P , so X_i \sim ...
is a limiting case of binomial random measure where
.
Distributional self-similarity applications
The "bone" condition on the pgf
of
encodes a distributional self-similarity property whereby all counts in restrictions (thinnings) to subspaces (encoded by pgf
) are in the same family as
of
through rescaling of the canonical parameter. These ideas appear closely connected to those of self-decomposability and stability of discrete random variables. Binomial thinning is a foundational model to count time-series. The
Poisson random measure Let (E, \mathcal A, \mu) be some measure space with \sigma-finite measure \mu. The Poisson random measure with intensity measure \mu is a family of random variables \_ defined on some probability space (\Omega, \mathcal F, \mathrm) such that
i) ...
has the well-known splitting property, is prototypical to the class of additive (completely random) random measures, and is related to the structure of
Levy processes, the jumps of
Kolmogorov equations (Markov jump process)
In mathematics and statistics, in the context of Markov processes, the Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, are a pair of systems of differential equations that describe the time evoluti ...
, and the excursions of
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
.
[Cinlar Erhan. Probability and Stochastics. Springer-Verlag New York; 2011.] Hence the self-similarity property of the PT family is fundamental to multiple areas. The PT family members are "primitives" or prototypical random measures by which many random measures and processes can be constructed.
References
{{reflist
Poisson distribution