How to identify a point process as Poisson point process with (possibly) random intensity measure?

Question

I got this question because of this post, discussing the limiting distribution of extreme point process of i.i.d. Gaussians seen from the tip. For notations please refer that post.

At first, @Iosif Pinelis gave an answer suggesting the number of points in a certain interval $[-A,0]$ is geometrically distributed, which sounds weird to me since the extreme point process seen from the critical speed is roughly Poisson point process, so how can it be so different when seen from the tip? At that time I just accepted it since the derivation is indeed correct.

But today I realized we can get the limiting distribution of the extreme point process seen from the tip, directly from the above Poisson point process, by noticing the random shift between these two point processes converges to some Gumbell distribution. This allows me to represent the geometric stuff by a Poisson point process with random intensity measure.

So it just came to me naturally:

Is it the case that any point process can be represented as a Poisson point process with a random intensity measure, if it satisfies the independence condition for PPP? (or do we need other necessary constraints?)

Suppose we have a description of a point process, like that @Iosif's geometric-type Laplace functional for $\bar{\mathcal{E}}$, the limit extreme point process seen from the tip, how to identify the random intensity measure to give that representation?

I may fail to organise my questions clearly, but basically I just find it extremely interesting that the geometric representation and the random intensity measure representation are actually the same. So I wonder whether it is universal, and if yes, practically how to achieve that.

Answer 1

$\bullet$ The answer to the first question is affirmative, but the notion of "independence" needs to be modified if we allow for the point process to be directed by an external random variable.

A Poisson point process with a random intensity is known as a Cox process. The "independence" property is defined in terms of a "thinning" procedure, see Lectures on the Poisson Process:

Given a point process $N$ on $\mathbb{R}^+$ and $p\in(0,1]$, the thinned process $N_p$ is obtained by retaining every point of the process with a probability $p$ and deleting it with probability $1-p$. Then $N$ is called the $p$-inverse of $N_p$.
Theorem: A point process is a Cox process if and only if, for every $p\in(0,1]$ there exists a point process which is the $p$-inverse of the process.

$\bullet$ Concerning the second question: For each integer $k$, the $k$-th factorial moment measure of the Cox process equals the $k$-the ordinary moment measure of the random intensity. So point process and random intensity have the same mean, and the variance of the random intensity equals the variance of the point process minus the mean.

This implies a simple necessary criterion for a point process to be Cox process: the variance must be greater than the mean ("bunching"). If the variance is smaller than the mean ("anti-bunching") the point process cannot be a Poisson process with a random intensity.