# Marked poisson process vs compounded

I am a bit fuzzy about difference between compounded poisson process defined as $$\sum_{i=1}^{N_t} D_i$$ where $N_t$ is poisson process and $D_i$ are iid random variables

and marked poisson process. Is compounded poisson a version of marked process $\{(\tau_i, D_i), i \in \mathbb{N} \}$ ?

• My opinion: A poisson counting process jumps up by one unit at random times. A compound poisson process is a generalization of this, which jumps up (or down) by a random amount, i.e. not always +1. The increments however are independent. In a marked poisson process, the "mark" has a certain prob distribution, but it is not necessarily equal to the previous mark plus/minus a random amount. The notion of "counting" or incrementatio is no longer there. Commented Nov 17, 2017 at 19:23

The compound Poisson process isn't technically a marked process because we formulate the process with respect to $$\sum_i D_i$$ instead of $$(\tau_i, D_i)$$. However the compound process is constructed from a marked process.

The compound Poisson point process or compound Poisson process is formed by adding random values or weights to each point of Poisson point process defined on some underlying space, so the process is constructed from a marked Poisson point process, where the marks form a collection of independent and identically distributed non-negative random variables.

https://en.wikipedia.org/wiki/Poisson_point_process#Compound_Poisson_point_process

The difference between a compound and marked Poisson process is that for a compound Poisson process the $$D_i$$ are non-negative iid but for a marked process the $$D_i$$ can depend all past history $$(\tau_i, D_{i-1}, \tau_{i-1}, ...)$$ and don't need to be non-negative or even a real number.

marks can be as diverse as integers, real numbers, lines, geometrical objects or other point processes.

https://en.wikipedia.org/wiki/Poisson_point_process#Marked_Poisson_point_process

As such, we can't really talk about $$\sum_i D_i$$ for a marked process because in general that sum isn't defined since $$D_i$$ might not even be a real number.

• Yes, nice answer. I want to emphasize how both the marked point process and the compounded Poisson process are semantically close but the difference boils down to how the two objects are defined. The former is more general but an easy and simplistic version is the later. Commented Jan 17 at 19:43