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.
be a non-negative integer-valued random variable
be a collection of iid random variables (stones) taking values in
depends on the pair of deterministic probability measures
through the stone throwing construction (STC) [5]
is a mixed binomial process[6] Let
is encoded in the Laplace functional
{\displaystyle \quad \mathbb {C} {\text{ov}}(Nf,Ng)=c\nu (fg)+(\delta ^{2}-c)\nu f\nu g}
The joint distribution of the collection
The following result extends construction of a random measure
represents some properties (marks) of
and the transition probability kernel
The probability law of the random measure is determined by its Laplace functional and hence generating function.
share the same family of laws subject to a rescaling
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.
belongs to the canonical non-negative power series (NNPS) family of distributions and
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
Poisson is additive with independence on disjoint sets, whereas negative binomial has positive covariance and binomial has negative covariance.
The "bone" condition on the pgf
encodes a distributional self-similarity property whereby all counts in restrictions (thinnings) to subspaces (encoded by pgf
through rescaling of the canonical parameter.
These ideas appear closely connected to those of self-decomposability and stability of discrete random variables.
[7] Binomial thinning is a foundational model to count time-series.
[8][9] The Poisson random measure has the well-known splitting property, is prototypical to the class of additive (completely random) random measures, and is related to the structure of Lévy processes, the jumps of Kolmogorov equations, and the excursions of Brownian motion.
[10] 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.