De Finetti-style theorem for Point Processes

Question

I am new to point processes. I know there are a number of theorems along the lines that if a point process $\eta$ satisfies:

1. Complete independence (the random variables $\eta(B_1), \ldots, \eta(B_n)$ are independent for pairwise disjoint bounded measurable $B_1, \ldots, B_n$) and

2. Some regularity conditions like being simple and uniform $\sigma$-finite on a Borel subset of a complete separable metric space,

then $\eta$ is a Poisson process.

It seems that something like the following must be known. I think (I haven't worked through all the details) that if $\eta$ satisfies:

1. (a) A condition one might call "complete exchangeability": for any disjoint bounded measurable $B_1, \ldots, B_n$, there are $A_i \subseteq B_i$, with equality for at least one $i$, such that $\eta(A_1), \ldots, \eta(A_n)$ are exchangeable random variables;

or equivalently

1. (b) For any disjoint bounded measurable $B_1, \ldots, B_n$ with $\mathbb{E}\eta(B_1) = \ldots = \mathbb{E}\eta(B_n)$, $\eta(B_1),\ldots, \eta(B_n)$ are exchangeable random variables;

as well as

2. Similar regularity conditions including $\mathbb{E}\eta(B)<\infty$ for all bounded measurable $B$; and
3. $\mathbb{E}\eta(\text{whole space}) = \infty$

then there exists a nonnegative-valued scalar random variable $G$ with $\mathbb{E}G = 1$ such that conditioned on $G$, $\eta$ is a Poisson process with intensity measure $G\mathbb{E}\eta$.

Without (3) there are simple counterexamples, e.g. $\eta = \delta_x$ where $x$ is distributed according to a nonatomic probability distribution.

Could anyone provide a reference for such a point process de Finetti theorem?

Answer 1

This is Theorem 3.34 in

Kallenberg, Olav, Random measures, Berlin: Akademie-Verlag. London - New York - San Francisco: Academic Press. 104 p. M 28.00 (1976). ZBL0345.60032.