Poisson-type random measure

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.[1] The PT family of distributions is also known as the Katz family of distributions,[2] the Panjer or (a,b,0) class of distributions[3] and may be retrieved through the Conway–Maxwell–Poisson distribution.[4]

Throwing stones

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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 (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) [5]

 

where   has law   and iid   have law  .   is a mixed binomial process[6]

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

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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

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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

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The probability law of the random measure is determined by its Laplace functional and hence generating function.

Definition: Bone

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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

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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, 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

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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.[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 (Markov jump process), 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.

References

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  1. ^ Caleb Bastian, Gregory Rempala. Throwing stones and collecting bones: Looking for Poisson-like random measures, Mathematical Methods in the Applied Sciences, 2020. doi:10.1002/mma.6224
  2. ^ Katz L.. Classical and Contagious Discrete Distributions ch. Unified treatment of a broad class of discrete probability distributions, :175-182. Pergamon Press, Oxford 1965.
  3. ^ Panjer Harry H.. Recursive Evaluation of a Family of Compound Distributions. 1981;12(1):22-26
  4. ^ Conway R. W., Maxwell W. L.. A Queuing Model with State Dependent Service Rates. Journal of Industrial Engineering. 1962;12.
  5. ^ Cinlar Erhan. Probability and Stochastics. Springer-Verlag New York; 2011
  6. ^ Kallenberg Olav. Random Measures, Theory and Applications. Springer; 2017
  7. ^ Steutel FW, Van Harn K. Discrete analogues of self-decomposability and stability. The Annals of Probability. 1979;:893–899.
  8. ^ Al-Osh M. A., Alzaid A. A.. First-order integer-valued autogressive (INAR(1)) process. Journal of Time Series Analysis. 1987;8(3):261–275.
  9. ^ Scotto Manuel G., Weiß Christian H., Gouveia Sónia. Thinning models in the analysis of integer-valued time series: a review. Statistical Modelling. 2015;15(6):590–618.
  10. ^ Cinlar Erhan. Probability and Stochastics. Springer-Verlag New York; 2011.