In this class, we have seen ways to describe the distribution of point processes, proved existence of Poisson processes, and defined non-Poisson processes through thinning, superposition, and randomizing the intensity measure of a Poisson process.
Today, we are going to go further in describing the distribution of finite point processes, and we will define a new class of point processes using densities with respect to a Poisson process.
Definition (Finite Point Processes). A point process \(\Phi\) on a CSMS \(E\) is finite if
\[
\Phi(E) < \infty \quad \text{almost surely}.
\]
Today we will consider \(E \subseteq \mathbb{R}^{d}\).
We have seen that a finite point process on \(E\) can be written as \[ \Phi = \sum_{i=1}^{N} \delta_{X_i}, \] where \(N\) is a random variable in \(\mathbb{N}_{0}\) and \(\{X_i\}\) are random elements of \(E\).
A full description of the distribution of \(\Phi\) can be obtained with:
We consider the points to be unordered, so we require that \(\Pi_n\) is symmetric, i.e., it should give equal weights to all \(n!\) permutations of the coordinates. If this is not already the case, we can take \(\Pi_n^{\mathrm{sym}}(A_1 \times \cdots \times A_n) = \frac{1}{n!} \sum_{\sigma} \Pi_n(A_{\sigma(1)} \times \cdots \times A_{\sigma(n)})\).
Definition (Janossy measure). For \(n \in \mathbb{N}\), the \(n\)-th Janossy measure of \(\Phi\) is the measure \(J_n\) on \(E^n\) defined by
\(J_n(B_1 \times \cdots \times B_n) = n! \, p_n \, \Pi_n^{\mathrm{sym}}(B_1 \times \cdots \times B_n)\),
for \(B_i \in \mathcal{B}(E)\), where \(\sum_{n=0}^{\infty} p_n = 1\) and \(p_n = \mathbb{P}(N = n)\).
In particular, this implies that for any measurable \(g : \mathcal{M}(E) \to \mathbb{R}_{+}\), \(\mathbb{E}[g(\Phi)] = g_0 p_0 + \sum_{n=1}^{\infty} \frac{1}{n!} \int_{E^n} g\!\big(\sum_{i=1}^{n} \delta_{x_i}\big)\, J_n(dx_1 \cdots dx_n)\), where \(g_0\) is the value of \(g\) for the zero measure.
If \(J_n\) is absolutely continuous with respect to Lebesgue measure, its density \(j_n\) is called the \(n\)-th Janossy density and it has the following interpretation:
\(j_n(x_1,\dots,x_n)\,dx_1 \cdots dx_n = \mathbb{P}\big(\text{there are exactly $n$ points of $\Phi$, one in each of the infinitesimal regions }(x_i,x_i+dx_i)\big)\).
Remark: Note that \(p_0 + \sum_{n=1}^{\infty} \frac{1}{n!} J_n(E^n) = 1\).
If we are given symmetric measures \(\{J_n\}_{n\ge 0}\) and \(p_0 \in [0,1]\) satisfying this relation, we can construct a finite point process \(\Phi\) on \(E\) with these Janossy measures. Indeed, we let
\(p_n = \frac{1}{n!} J_n(E^n) = \mathbb{P}(N = n)\),
and if \(p_n > 0\), define
\(\Pi_n(B) = \frac{J_n(B)}{J_n(E^n)}\) for \(B \in \mathcal{B}(E^n)\).
Let \(\Phi\) be a finite point process on \(E \subseteq \mathbb{R}^d\).
Question: How do Janossy measures relate to other descriptors of the distribution of \(\Phi\)?
Proposition. For any disjoint \(B_1,\dots,B_k \in \mathcal{B}(E)\) and integers \(n_1,\dots,n_k\), \(\mathbb{P}[\Phi(B_1)=n_1,\dots,\Phi(B_k)=n_k] = \frac{1}{n_1!\cdots n_k!} \sum_{r \in \mathbb{N}_0} J_{n+r}(B_1^{n_1} \times \cdots \times B_k^{n_k} \times C^{r})\), where \(C = E \setminus (B_1 \cup \cdots \cup B_k)\) and \(n = \sum_{i=1}^{k} n_i\).
Here one can write this quantity as \(\sum_{r \in \mathbb{N}_0} \mathbb{P}\big(\Phi(B_1)=n_1,\dots,\Phi(B_k)=n_k,\ \Phi(E)=n+r\big)\), highlighting the contribution from configurations with total count \(n+r\). See Corollary 4.39 in [baccelli2024random] for detailed proof.
Recall: The \(k\)-th factorial moment measure \(\alpha^{(k)}\) of \(\Phi\) on \(E^k\) is defined by
\(\alpha^{(k)}(B) = \mathbb{E}[\Phi^{(k)}(B)]\) for \(B \in \mathcal{B}(E^k)\),
where \(\Phi^{(k)} = \sum_{i_1 \neq \cdots \neq i_k} \delta_{(X_{i_1},\dots,X_{i_k})}\).
Proposition. For \(B \in \mathcal{B}(E^k)\), \(\alpha^{(k)}(B) = \sum_{n \in \mathbb{N}_0} \frac{1}{n!} J_{k+n}(B \times E^n)\).
See Proposition 4.3.10 in [baccelli2024random] for the proof.
Recall that if \(\alpha^{(k)}\) is absolutely continuous with respect to Lebesgue measure, its density \(\rho^{(k)}\) is called the \(k\)-th product density of \(\Phi\) and has the interpretation
\(\rho^{(k)}(x_1,\dots,x_k)\,dx_1 \cdots dx_k = \mathbb{P}\big(\text{there is a point of $\Phi$ in each }(x_i,x_i+dx_i)\big)\).
If product and Janossy densities exist, then \[\rho^{(k)}(x_1,\dots,x_k) = \sum_{n=0}^{\infty} \frac{1}{n!} \int_{E^n} j_{k+n}(x_1,\dots,x_k,y_1,\dots,y_n)\,dy_1\cdots dy_n\].
Example. The Janossy measures of a Poisson process on \(E\) with intensity measure \(\Lambda\) are
\(J_n(\cdot) = e^{-\Lambda(E)}\,\Lambda^{n}(\cdot)\).