# 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. The PT family of distributions is also known as the Katz family of distributions, the Panjer or (a,b,0) class of distributions and may be retrieved through the Conway–Maxwell–Poisson distribution. A major contributor to this article appears to have a close connection with its subject.(May 2020)

## . . . Poisson-type random measure . . .

Let

${displaystyle K}$ be a non-negative integer-valued random variable

${displaystyle Kin mathbb {N} _{geq 0}=mathbb {N} _{>0}cup {0}}$ ) with law

${displaystyle kappa }$ , mean

${displaystyle cin (0,infty )}$ and when it exists variance

${displaystyle delta ^{2}>0}$ . Let

${displaystyle nu }$ be a probability measure on the measurable space

${displaystyle (E,{mathcal {E}})}$ . Let

${displaystyle mathbf {X} ={X_{i}}}$ be a collection of iid random variables (stones) taking values in

${displaystyle (E,{mathcal {E}})}$ with law

${displaystyle nu }$ .

The random counting measure

${displaystyle N}$ on

${displaystyle (E,{mathcal {E}})}$ depends on the pair of deterministic probability measures

${displaystyle (kappa ,nu )}$ through the stone throwing construction (STC) 

${displaystyle quad N_{omega }(A)=N(omega ,A)=sum _{i=1}^{K(omega )}mathbb {I} _{A}(X_{i}(omega ))quad {text{for}}quad omega in Omega ,,,,Ain {mathcal {E}}}$ where

${displaystyle K}$ has law

${displaystyle kappa }$ and iid

${displaystyle X_{1},X_{2},dotsb }$ have law

${displaystyle nu }$ .

${displaystyle N}$ is a mixed binomial process

Let

${displaystyle {mathcal {E}}_{+}={f:Emapsto mathbb {R} _{+}}}$ be the collection of positive

${displaystyle {mathcal {E}}}$ -measurable functions. The probability law of

${displaystyle N}$ is encoded in the Laplace functional

${displaystyle quad mathbb {E} e^{-Nf}=mathbb {E} (mathbb {E} e^{-f(X)})^{K}=mathbb {E} (nu e^{-f})^{K}=psi (nu e^{-f})quad {text{for}}quad fin {mathcal {E}}_{+}}$ where

${displaystyle psi (cdot )}$ is the generating function of

${displaystyle K}$ . The mean and variance are given by

${displaystyle quad mathbb {E} Nf=cnu f}$ and

${displaystyle quad mathbb {V} {text{ar}}Nf=cnu f^{2}+(delta ^{2}-c)(nu f)^{2}}$ The covariance for arbitrary

${displaystyle f,gin {mathcal {E}}_{+}}$ is given by

${displaystyle quad mathbb {C} {text{ov}}(Nf,Ng)=cnu (fg)+(delta ^{2}-c)nu fnu g}$ When

${displaystyle K}$ is Poisson, negative binomial, or binomial, it is said to be Poisson-type (PT). The joint distribution of the collection

${displaystyle N(A),ldots ,N(B)}$ is for

${displaystyle i,ldots ,jin mathbb {N} }$ and

${displaystyle i+cdots +j=k}$ ${displaystyle mathbb {P} (N(A)=i,ldots ,N(B)=j)=mathbb {P} (N(A)=i,ldots ,N(B)=j|K=k),mathbb {P} (K=k)={frac {k!}{i!cdots j!}},nu (A)^{i}cdots nu (B)^{j},mathbb {P} (K=k)}$ The following result extends construction of a random measure

${displaystyle N=(kappa ,nu )}$ to the case when the collection

${displaystyle mathbf {X} }$ is expanded to

${displaystyle (mathbf {X} ,mathbf {Y} )={(X_{i},Y_{i})}}$ where

${displaystyle Y_{i}}$ is a random transformation of

${displaystyle X_{i}}$ . Heuristically,

${displaystyle Y_{i}}$ represents some properties (marks) of

${displaystyle X_{i}}$ . We assume that the conditional law of

${displaystyle Y}$ follows some transition kernel according to

${displaystyle mathbb {P} (Yin B|X=x)=Q(x,B)}$

.