Spatial Skellam-Type Point Process

• Spatial Skellam-type point processes are defined via generalized Poisson random fields that model clustered and overdispersed event occurrences in spatial domains.
• They employ closed-form probability mass and generating functions, enabling tractable inference and exact, efficient sampling methods.
• The flexible structure supports applications in ecology, telecommunications, and epidemiology by effectively modeling multiple events and rich dependence patterns.

A Generalized Poisson Random Field (GPRF) is a family of discrete random measures and multivariate processes that generalize classical Poisson random fields and processes. GPRFs capture flexible dependence and allow overdispersion, cluster structures, and richer event-size multiplicity than classical Poisson fields. Their definition encompasses point process models in spatial domains, tree-structured Markov random fields for multivariate discrete data, and fractional extensions via random time changes. GPRFs admit closed-form probability mass functions (pmf), probability generating functions (pgf), and exact sampling algorithms. Their formulations enable tractable inference, flexible aggregation, and quantification of dependence structures in high-dimensional count data and spatial systems.

1. Fundamental Definitions and Constructions

A GPRF is rigorously defined as a non-negative, integer-valued random measure $M$ on a domain $A \subset \mathbb{R}^d$, characterized by a multiparameter family ${\lambda_j>0}_{j=1}^k$ to model the intensity of point masses (or event-sizes) $j$ within $A$ (Vishwakarma, 23 Jan 2026). For rectangles $$A \in \mathcal A_d = \{[0,x] : x \in \mathbb{R}^d_+, |A|<\infty\}$$, the law is

$$\Pr\{M(A)=n\} =\sum_{(n_1,\dots,n_k)\in\Theta(k,n)} \prod_{j=1}^k \frac{(\lambda_j |A|)^{n_j}}{n_j!} e^{-\lambda_j|A|}, \quad n \ge 0,$$

with $$\Theta(k, n) = \{(n_1, ..., n_k)\in\mathbb N_0^k : \sum_{j=1}^k j n_j = n\}$$. The GPRF process assigns probability mass not only to single points but also to event clusters (with multiplicities), allowing richer modeling of spatial phenomena such as clumped points or batch arrivals.

On a tree-structured Markov random field, a multivariate GPRF $N = (N_v)_{v \in V}$ is constructed via binomial-thinning recursions over the tree topology, with marginal distributions $N_v \sim \mathrm{Poisson}(\lambda)$ for all $v$ and $\lambda > 0$ decoupled from edge-dependence parameters $\alpha_e \in [0, 1]$. The generative recursion (Côté et al., 2024):

$$N_r = L_r,\quad N_v = [\alpha_{(\mathrm{pa}(v), v)} \circ N_{\mathrm{pa}(v)}] + L_v,\quad v \neq r,$$

where $$L_v \sim \mathrm{Poisson}(\lambda(1 - \alpha_{(\mathrm{pa}(v), v)}))$$ and $\alpha \circ X$ denotes binomial thinning, realizes a model with Poisson marginals for all nodes, local Markov property, and scalable dependence control.

2. Joint Probability Laws and Generating Functions

For spatial GPRFs, the pgf over $A$ is

$$G(z;A) = \mathbb{E}[z^{M(A)}] = \exp \left\{ \sum_{j=1}^k \lambda_j |A| (z^j - 1) \right\},$$

yielding mean and variance

$$\mathbb{E} M(A) = \sum_{j=1}^k j\,\lambda_j\,|A|, \qquad \mathrm{Var}(M(A)) = \sum_{j=1}^k j^2\,\lambda_j\,|A|.$$

Covariance for overlaps $A,B$ is $\sum_{j=1}^k j^2 \lambda_j |A \cap B|$ (Vishwakarma, 23 Jan 2026).

For tree-structured GPRFs, explicit joint pmf expressions are available (Côté et al., 2024):

$$\Pr\{N = x\} = e^{-\lambda} \frac{\lambda^{x_r}}{x_r!} \prod_{v \neq r} \sum_{k=0}^{\min(x_{\mathrm{pa}(v)}, x_v)} \binom{x_{\mathrm{pa}(v)}}{k} \alpha_{(\mathrm{pa}(v), v)}^k (1 - \alpha_{(\mathrm{pa}(v), v)})^{x_{\mathrm{pa}(v)} - k} e^{-\lambda(1 - \alpha_{(\mathrm{pa}(v), v)})} \frac{[\lambda(1 - \alpha_{(\mathrm{pa}(v), v)})]^{x_v - k}}{(x_v - k)!}.$$

A recursive construction for the multivariate pgf incorporates the dependence structure via a mapping $\eta_v$ over each node and its descendants.

Table: Example pmfs for GPRF variants

Model Type	PMF Formula (Summary)	Reference
Spatial GPRF ($k$ types)	$$\sum_{\Theta(k,n)} \prod_j \mathrm{Poisson}(n_j; \lambda_j|A|)$$	(Vishwakarma, 23 Jan 2026)
Tree GPRF	Factorization involving Poisson parent, binomial thinning	(Côté et al., 2024)
Fractional GPRF	Involves Mittag-Leffler and Wright functions, time-changed pgf	(Kataria et al., 2024)

3. Compound-Poisson and Thinning Representations

A GPRF admits compound Poisson and superposition representations (Vishwakarma, 23 Jan 2026). Specifically, $M$ is a GPRF with parameters ${\lambda_j}$ if and only if there exist independent classical Poisson fields $N_j$ of rates $\lambda_j$ so that

$M(A) \overset{d}{=} \sum_{j=1}^k j\,N_j(A).$

Equivalently, $M$ can be viewed as a sum over a Poisson field $N$ of rate $\Lambda = \sum_{j=1}^k \lambda_j$, with iid random marks $X_r$ (taking values $j=1,...,k$ with $\Pr\{X_r=j\}= \lambda_j/\Lambda$).

Thinning rules generalize: classical Poisson thinning yields independent subfields with reduced rates; GPRF-thinning applies independent thinning operations by event-size, preserving independence for thinned subfields (Vishwakarma, 23 Jan 2026).

4. Fractional and Time-Changed Generalizations

Fractional GPRFs are constructed via random time changes employing inverse-stable subordinators (Kataria et al., 2024, Vishwakarma, 23 Jan 2026). For parameters $0<\alpha,\beta<1$, the FGPRF on $\mathbb{R}^2_+$ is realized as

$$M^{\alpha,\beta}(s,t) = M(L^\alpha(s), L^\beta(t)),$$

where $L^\alpha$ and $L^\beta$ are independent stochastic processes with heavy-tailed waiting times. The discrete laws, means, variances, and covariance structures are governed by fractional partial differential equations and expressed in terms of generalized Wright and three-parameter Mittag-Leffler functions.

In general dimension $d\ge 1$, for volume $|B|$ and parameters $\lambda, \alpha, \gamma$, the pmf is

$$p_{α,γ}(k;B) = \frac{(\gamma)_k [\lambda |B|^\alpha]^k}{k!} E_{α, \alpha k+1}^{\gamma+k}(-\lambda |B|^\alpha),$$

with corresponding pgf and mean/variance relations involving $\Gamma$ functions and fractional calculus (Kataria et al., 2024).

5. Dependence Structures and Marginal-Decoupling

A distinctive property—evident in tree-structured GPRFs—is the decoupling of marginal rates $\lambda$ from dependence parameters $\alpha_e$ (Côté et al., 2024). Marginal means and variances are solely governed by $\lambda$, independently of the dependence graph topology or the strength parameters $\alpha_e$. This property is rare among graphical count models, wherein typically, marginal laws intertwine with dependence coefficients. This separation facilitates tractable inference: one can calibrate intensity surfaces ($\lambda$) independently from adjusting edge correlations ($\alpha_e$).

Stochastic ordering results show that raising $\alpha_e$ (holding other parameters fixed) strengthens supermodular ordering and yields convex ordering of aggregated sums. The limiting cases $\alpha \equiv 0$ and $\alpha \equiv 1$ correspond to fully independent Poisson marginals and comonotonic Poisson vectors, respectively.

6. Sampling Procedures and Computational Advantages

GPRFs, especially tree-structured variants, admit single-pass exact samplers with $O(n)$ computational complexity—$n$ Poisson draws plus $n-1$ binomial draws per sample (Côté et al., 2024). This enables scalable simulation of high-dimensional discrete models.

For spatial and fractional GPRFs, closed-form expressions for pmf and pgf, as well as path-integral statistics, allow efficient aggregation (e.g., fast Fourier transform, Panjer-type recursions) and tractable allocation formulas for functionals such as $E[X_i \mathbb{1}_{S=k}]$ (Côté et al., 2024, Vishwakarma, 23 Jan 2026, Kataria et al., 2024).

7. Applications, Extensions, and Comparative Context

GPRFs generalize classical Poisson random fields, accommodating multiplicities in infinitesimal regions and modeling overdispersion or clustered patterns robustly (Vishwakarma, 23 Jan 2026). Applications span ecology (species clumping), telecommunications (batch arrivals), epidemiology (multiple infections per region), and materials science (clustered defects). Fractional and compound extensions further enable modeling of long-range correlations and anomalous/heterogeneous propagation phenomena (Kataria et al., 2024).

GPRF-based models can yield exact Poisson marginals with nugget-free covariances, mean-square continuity, and composite-likelihood inference via explicit bivariate pmfs (Morales-Navarrete et al., 2021). Empirical applications demonstrate competitive zero-inflated extensions (ZIP-GPRF), outperforming alternatives such as Poisson Log-Gaussian (Poisson-LG) and Poisson Gaussian copula (Poisson-GC) models in fit and tractability.

Generalized Skellam-type point processes defined via GPRFs, as well as fractional Skellam fields, admit compound Poisson representations and closed-form moment-generating functions (mgf), extending the operational calculus of spatial point process models (Vishwakarma, 23 Jan 2026).

Papers of record include Di Crescenzo et al. (2016), Vishwakarma–Kataria (2025), Morales-Navarrete et al. (2021), Kataria–Vishwakarma (2024), and recent foundational works (Côté et al., 2024, Vishwakarma, 23 Jan 2026, Morales-Navarrete et al., 2021, Kataria et al., 2024).