In most of the literature, I come across marked Poisson Poisson process with just one marks. Can someone give me details on Marked Poisson process with multiple marks?

## 1 Answer

I have been interested in the same question recently and, just as you, I did not find much in literature. What kind of details do you need?

Here I report a few basic properties one can easily find, as a starting point.

First denote with $\lambda$ the exponential rate of the process and with $\varphi$ the probability distribution of marks (we assume that marks can only be positive). Consider the marked Poisson process on the line and define the random variable $\mathcal{N}(a,b)$, that counts the number of marks in the interval $(a,b)$. The probability law of $\mathcal{N}(a,b)$ is \begin{equation} \mathbb{P}(\mathcal{N}(a,b)=N)=\sum_{\mu \vdash N} \sum_{\eta \sim \mu} e^{-\lambda(b-a)} \frac{\lambda^{l(\mu)}(b-a)^{l(\mu)}}{l(\mu)!}\varphi(\eta_1) \cdots \varphi(\eta_{l(\mu)}), \end{equation} where the symbol $\mu \vdash N$ means that $\mu$ is a partition of $N$, the symbol $\eta \sim \mu$ means that $\eta$ is a permutation of $\mu$ and $l(\mu)$ is the length of the partition $\mu$.

We can also compute the generating function of $\mathcal{N}(a,b)$ as \begin{equation} \mathbb{E}(x^{\mathcal{N}(b-a)})=\exp\{ \lambda(b-a) (\mathbb{E}_\varphi(x^{\mathfrak{m}}) -1) \}, \end{equation} where $\mathfrak{m}$ is the number of marks at a specific location.

What other remarkable quantities do you think are interesting? (For example I have been interested in an inhomogeneous version of the marked Poisson process)

NB. Originally this was supposed to be just a comment, bei since my score here is low I was forced to post it as an answer. I hope you don't mind.