Poisson-type random measure

Let $K$ be a non-negative integer-valued random variable $K\backslash in\backslash mathbb\{N\}\_\{\backslash ge0\}=\backslash mathbb\{N\}\_\{>0\}\backslash cup\backslash \{0\backslash \}$) with law $\backslash kappa$, mean $c\backslash in(0,\backslash infty)$ and when it exists variance $\backslash delta^2>0$. Let $\backslash nu$ be a probability measure on the measurable space $(E,\backslash mathcal\{E\})$. Let $\backslash mathbf\{X\}=\backslash \{X\_i\backslash \}$ be a collection of iid random variables (stones) taking values in $(E,\backslash mathcal\{E\})$ with law $\backslash nu$.

The random counting measure $N$ on $(E,\backslash mathcal\{E\})$ depends on the pair of deterministic probability measures $(\backslash kappa,\backslash nu)$ through the stone throwing construction (STC) Cinlar Erhan. Probability and Stochastics. Springer-Verlag New York; 2011

$\backslash quad\; N\_\backslash omega(A)\; =\; N(\backslash omega,A)\; =\; \backslash sum\_\{i=1\}^\{K(\backslash omega)\}\backslash mathbb\{I\}\_A(X\_i(\backslash omega))\backslash quad\; \backslash text\{for\}\; \backslash quad\backslash omega\backslash in\backslash Omega,\backslash ,\backslash ,\backslash ,A\backslash in\backslash mathcal\{E\}$

where $K$ has law $\backslash kappa$ and iid $X\_1,X\_2,\backslash dotsb$ have law $\backslash nu$. $N$ is a mixed binomial processKallenberg Olav. Random Measures, Theory and Applications. Springer; 2017

Let $\backslash mathcal\{E\}\_+=\backslash \{f:\; E\backslash mapsto\backslash mathbb\{R\}\_+\backslash \}$ be the collection of positive $\backslash mathcal\{E\}$-measurable functions. The probability law of $N$ is encoded in the Laplace functional

$\backslash quad\backslash mathbb\{E\}\; e^\{-N\; f\}\; =\backslash mathbb\{E\}\; (\backslash mathbb\{E\}\; e^\{-f(X)\})^K\; =\backslash mathbb\{E\}\; (\backslash nu\; e^\{-f\})^K=\backslash psi(\backslash nu\; e^\{-f\})\backslash quad\backslash text\{for\}\backslash quad\; f\backslash in\backslash mathcal\{E\}\_+$

where $\backslash psi(\backslash cdot)$ is the generating function of $K$. The mean and variance are given by

$\backslash quad\backslash mathbb\{E\}\; Nf\; =c\backslash nu\; f$

and

$\backslash quad\backslash mathbb\{V\}\backslash text\{ar\}\; Nf\; =\; c\backslash nu\; f^2\; +\; (\backslash delta^2-c)\; (\backslash nu\; f)^2$

The covariance for arbitrary $f,g\backslash in\backslash mathcal\{E\}\_+$ is given by

$\backslash quad\backslash mathbb\{C\}\backslash text\{ov\}(Nf,Ng)\; =\; c\backslash nu(fg)\; +\; (\backslash delta^2-c)\backslash nu\; f\; \backslash nu\; g$

When $K$ is Poisson, negative binomial, or binomial, it is said to be Poisson-type (PT). The joint distribution of the collection $N(A),\backslash ldots,N(B)$ is for $i,\backslash ldots,\; j\; \backslash in\; \backslash N$ and $i+\backslash cdots+j\; =k$

: $$

\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 $N=(\backslash kappa,\backslash nu)$ to the case when the collection $\backslash mathbf\{X\}$ is expanded to $(\backslash mathbf\{X\},\backslash mathbf\{Y\})=\backslash \{(X\_i,Y\_i)\backslash \}$ where $Y\_i$ is a random transformation of $X\_i$. Heuristically, $Y\_i$ represents some properties (marks) of $X\_i$. We assume that the conditional law of $Y$ follows some transition kernel according to $\backslash mathbb\{P\}(Y\backslash in\; B|X=x)=Q(x,B)$.

Consider random measure $N=(\backslash kappa,\backslash nu)$ and the transition probability kernel $Q$ from $(E,\; \backslash cal\; E)$ into $(F,\; \backslash cal\; F)$. Assume that given the collection $\backslash mathbf\{X\}$ the variables $\backslash mathbf\{Y\}=\backslash \{Y\_i\backslash \}$ are conditionally independent with $Y\_i\backslash sim\; Q(X\_i,\backslash cdot)$. Then $M=(\backslash kappa,\; \backslash nu\backslash times\; Q)$ is a random measure on $(E\backslash times\; F,\; \backslash cal\; E\backslash otimes\; F)$. Here $\backslash mu=\backslash nu\backslash times\; Q$ is understood as $\backslash mu(dx,dy)=\backslash nu(dx)Q(x,dy)$. Moreover, for any $f\backslash in\; (\{\backslash cal\; E\}\backslash otimes\; \{\backslash cal\; F\})\_+$ we have that $\backslash mathbb\{E\}\; e^\{-M\; f\}=\backslash psi(\backslash nu\; e^\{-g\})$ where $\backslash psi(\backslash cdot\; )$ is pgf of $K$ and $g\backslash in\; \backslash mathcal\{E\}\_+$ is defined as $e^\{-g(x)\}=\; \backslash int\_F\; Q(x,dy)e^\{-f(x,y)\}.$

The following corollary is an immediate consequence.

The quantity $N\_A=(N\backslash mathbb\{I\}\_A,\backslash nu\_A)$ is a well-defined random measure on the measurable subspace $(E\backslash cap\; A,\; \backslash mathcal\{E\}\_A)$ where $\backslash mathcal\{E\}\_A=\backslash \{A\backslash cap\; B:\; B\backslash in\backslash mathcal\{E\}\backslash \}$ and $\backslash nu\_A(B)=\backslash nu(A\backslash cap\; B)/\backslash nu(A)$. Moreover, for any $f\backslash in\backslash mathcal\{E\}\_+$, we have that $\backslash mathbb\{E\}\; e^\{-N\_A\; f\}\; =\; \backslash psi(\backslash nu\; e^\{-f\}\backslash mathbb\{I\}\_A+b)$ where $b=1-\backslash nu(A)$.

Note $\backslash psi(\backslash nu\; e^\{-f\}\backslash mathbb\{I\}\_A+1-a)=\backslash psi\_A(\backslash nu\_A\; e^\{-f\})$ where we use $\backslash nu\; e^\{-f\}\backslash mathbb\{I\}\_A=a\backslash nu\_A\; e^\{-f\}$.

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

Let $K\_A\; =\; N\backslash mathbb\{I\}\_A$ be the counting variable of $K$ restricted to $A\backslash subset\; E$. When $\backslash \{N\backslash mathbb\{I\}\_A:\; A\backslash subset\; E\backslash \}$ and $K=N\backslash mathbb\{I\}\_E$ share the same family of laws subject to a rescaling $h\_a(\backslash theta)$ of the parameter $\backslash theta$, then $K$ is a called a bone distribution. The bone condition for the pgf is given by

$\backslash psi\_\{\backslash theta\}(at+1-a)=\backslash psi\_\{h\_a(\backslash theta)\}(t)$.

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.

Assume that $K\backslash sim\; \backslash kappa\_\backslash theta$ with pgf $\backslash psi\_\backslash theta$ belongs to the canonical non-negative power series (NNPS) family of distributions and $\backslash \{0,1\backslash \}\backslash subset\backslash text\{supp\}(K)$. Consider the random measure $N=(\backslash kappa\_\backslash theta,\backslash nu)$ on the space $(E,\backslash mathcal\{E\})$ and assume that $\backslash nu$ is diffuse. Then for any $A\backslash subset\; E$ with $\backslash nu(A)=a>0$ there exists a mapping $h\_a:\backslash Theta\backslash rightarrow\backslash Theta$ such that the restricted random measure is $N\_A=(\backslash kappa\_\{h\_a(\backslash theta)\},\backslash nu\_A)$, that is,

$\backslash quad\; \backslash mathbb\{E\}\; e^\{-N\_A\; f\}\; =\; \backslash psi\_\{h\_a(\backslash theta)\}(\backslash nu\_A\; e^\{-f\})\backslash quad\; \backslash text\{for\}\backslash quad\; f\backslash in\backslash mathcal\{E\}\_+$

iff $K$ 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 $N\backslash mathbb\{I\}\_A$ share the same family of distribution as $K$, 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 $p\backslash rightarrow\; 1,\; n\backslash rightarrow\; c$.

The "bone" condition on the pgf $\backslash psi\_\backslash theta$ of $K$ encodes a distributional self-similarity property whereby all counts in restrictions (thinnings) to subspaces (encoded by pgf $\backslash psi\_A$) are in the same family as $\backslash psi\_\backslash theta$ of $K$ through rescaling of the canonical parameter. These ideas appear closely connected to those of self-decomposability and stability of discrete random variables.Steutel FW, Van Harn K. Discrete analogues of self-decomposability and stability. The Annals of Probability. 1979;:893–899. Binomial thinning is a foundational model to count time-series.Al-Osh M. A., Alzaid A. A.. First-order integer-valued autoregressive (INAR(1)) process. Journal of Time Series Analysis. 1987;8(3):261–275.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. 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.Cinlar Erhan. Probability and Stochastics. Springer-Verlag New York; 2011. 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.

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Category:Poisson distribution