We now introduce the concept of higher moment measures, which play a key role in characterizing the distributional structure of a point process. We denote by $ E^n = E E (n ) $. The product \(\sigma\)-algebra on \(E^n\) is written as \[ \mathcal{B}(E^n) := \text{product } \sigma\text{-algebra generated by } \{ B_1 \times \dots \times B_n \,;\ B_i \in \mathcal{B}(E) \}. \]
Definition (Power of a counting measure). Let \(\nu \in \mathcal{M}(E)\) be given by $ = {i=1}^k {x_i}, k _0 {}. $ Then, the \(n\)-th power of \(\nu\) is the counting measure \(\nu^n\) on \((E^n, \mathcal{B}(E^n))\) defined by \[ \nu^n = \sum_{i_1, \dots, i_n \leq k} \delta_{(x_{i_1}, \dots, x_{i_n})}. \]
Intuitively, \(\nu^n\) counts all ordered \(n\)-tuples of points from the support of \(\nu\). In particular, for measurable sets \(B_1, \dots, B_n \in \mathcal{B}(E)\), we have \[ \nu^n(B_1 \times \dots \times B_n) = \sum_{i_1, \dots, i_n \leq k} \mathbf{1}\{ x_{i_1} \in B_1, \dots, x_{i_n} \in B_n \} = \nu(B_1) \cdots \nu(B_n). \]
Definition (\(n\)-th moment measure). The \(n\)-th moment measure of a point process \(\Phi\), denoted by \(\mu^{(n)}\), is the intensity measure of \(\Phi^n\).
That is, for any measurable sets \(B_1, \dots, B_n \in \mathcal{B}\), \[ \mu^{(n)}(B_1 \times \dots \times B_n) = \mathbb{E}\big[ \Phi(B_1) \cdots \Phi(B_n) \big]. \]
Definition (Factorial power of a counting measure). Let \(\nu \in \mathcal{M}(E)\) such that $ = {i=1}^k {x_i}, k _0 {}. $
The \(n\)-th factorial power of \(\nu\) is the counting measure \(\nu^{(n)}\) on \(E^n\) defined by \[ \nu^{(n)} = \sum_{i_1, \dots, i_n \leq k}^{\neq} \delta_{(x_{i_1}, \dots, x_{i_n})}, \] where the summation is taken over all \(n\)-tuples of distinct points of \(\nu\).
Example. If \(B_1, \dots, B_n\) are pairwise disjoint subsets of \(E\), then \[ \nu^{(n)}(B_1 \times \dots \times B_n) = \nu^n(B_1 \times \dots \times B_n). \]
In particular, \[ \nu^{(n)}(B^n) = \nu(B) (\nu(B)-1) \cdots (\nu(B) - n + 1), \] and for the second order case, \[ \nu^{(2)}(A \times B) = \nu(A)\nu(B) - \nu(A \cap B). \]
Definition (Factorial moment measure). The \(n\)-th factorial moment measure of a point process \(\Phi\), denoted by \(\alpha^{(n)}\), is defined as the intensity measure of \(\Phi^{(n)}\) on \(E^n\).
Example (Second order moments). The variance of the number of points in a set \(B\) is \[ \mathrm{Var}[\Phi(B)] = \mu^{(2)}(B \times B) - \Lambda(B)^2 = \alpha^{(2)}(B \times B) + \Lambda(B) - \Lambda(B)^2. \]
The covariance between \(\Phi(A)\) and \(\Phi(B)\) is \[ \mathrm{Cov}[\Phi(A), \Phi(B)] = \mu^{(2)}(A \times B) - \Lambda(A)\Lambda(B) = \alpha^{(2)}(A \times B) + \Lambda(A \cap B) - \Lambda(A)\Lambda(B). \]
Finally, applying Campbell’s theorem to \(\Phi^n\) and \(\Phi^{(n)}\), we have: \[ \mathbb{E}\left[ \sum_{(x_1, \dots, x_n) \in \Phi^n} f(x_1, \dots, x_n) \right] = \int_{E^n} f(x_1, \dots, x_n) \, d\mu^{(n)}(x_1, \dots, x_n), \] and \[ \mathbb{E}\left[ \sum_{(x_1, \dots, x_n) \in \Phi^{(n)}} f(x_1, \dots, x_n) \right] = \int_{E^n} f(x_1, \dots, x_n) \, d\alpha^{(n)}(x_1, \dots, x_n). \]