# 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]

 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) [5]

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

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[6]

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

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

.