What is a "Gaussian" point process?

Question

I try to understand the definition of a "Gaussian point process" which is (vaguely) given in this paper in section 2.2.1.

Let $(E,\mathcal E)$ be a measurable space. A random measure on $(E,\mathcal E)$ is a transition kernel from a probability space $(\Omega,\mathcal A,\operatorname P)$ to $(E,\mathcal E)$. A point process $\kappa$ on $(E,\mathcal E)$ is a random measure on $(E,\mathcal E)$ such that $\kappa(\omega,\;\cdot\;)$ is a counting measure for $\operatorname P$-almost all $\omega\in\Omega$. Since they don't give such a definition in the paper, they might assume that $$\kappa(\omega,\;\cdot\;)=\sum_{i\in I}\delta_{X_i(\omega)}\;\;\;\text{for all }\omega\in\Omega\tag1$$ for some countable set $I$ and an $(E,\mathcal E)$-valued process $(X_i)_{i\in I}$ on $(\Omega,\mathcal A,\operatorname P)$. In that case, we need to assume $\{x\}\in\mathcal E$ for all $x\in E$.

Now, they assume $E=\mathbb R^d$ for some $d\in\mathbb N$ and "define" that the point process is Gaussian if for every $k\in\mathbb N$ and $B_1,\ldots,B_k\in\mathcal B(\mathbb R^d)$, the random variable $(\kappa(\;\cdot\;,B_1),\ldots,\kappa(\;\cdot\;,B_k))$ has a multivariate Gaussian distribution.

My problem with this is: Is that possible at all? I mean, each $\kappa(\;\cdot\;,B_i)$ is $\mathbb N_0\cup\{\infty\}$-valued ... So, can it be normally distributed? I think the answer should be no ... So, what am I missing?

EDIT: If this actually does not make sense, can we fix the definition?

Answer 1

Of course formally we can't use gaussian distribution in case when r.v. take values in $\mathbb{Z}$. But nevertheless there's a sense when we use gaussian distribution in such situations. Let's prove it.

1. According to CLT sum of discrete r.v. $\approx$ normal, even though the sum of discrete r.v. is discrete.

2. In physics we say that errors are gaussian, but in fact they are discrete (because the measuring device has a scale with a finite number of divisions).

3. In biology we use normal distribution for nonnegative values such as weight of animals. But of course nondegenerate gaussian r.v. can't take values only in $\{ x| x > 0 \}$.

So gaussian point process $X$ looks like a limit of some sequence $x_n$ of "usual" point process and is smth.like $X_{n_0}$ with 'big' $n_0$.

And the assumption "the number of points is gaussian" is not more strange than the assumption "errors of measuring device are gaussian".