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, negative binomial distribution, and binomial distribution. The PT family of distributions is also known as the Katz family of distributions, the Panjer or class of distributions and may be retrieved through the Conway–Maxwell–Poisson distribution.
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. Let be a collection of iid random variables taking values in with law. The random counting measure on depends on the pair of deterministic probability measures through the stone throwing construction 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. 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 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 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 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. 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, 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 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 to subspaces 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 has the well-known splitting property, is prototypical to the class of additive random measures, and is related to the structure of Levy processes, the jumps of Kolmogorov equations, and the excursions of Brownian motion. 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.
