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Some Spatial Point Processes of Poisson Family

Pradeep Vishwakarma

TL;DR

This paper advances the theory of spatial Poisson-type processes by introducing the generalized Poisson random field (GPRF) on $\mathbb{R}^d_+$ and establishing its representation as a weighted sum of independent Poisson random fields, along with a complete infinitesimal characterization, capacity functional, and thinning results. It extends GPRF to the positive quadrant as a two-parameter Lévy process with rectangular increments, develops integrals and fractional variants via inverse stable subordination, and derives governing fractional differential equations and Wright-function based expressions for the FGPRF. Building on these foundations, the work defines a generalized Skellam point process (GSPP) from independent GPRFs and studies its fractional counterpart, including a compound Poisson field representation and explicit finite-dimensional distributions. The framework unifies 2D time-changed Poisson models, provides exact and fractional dynamics for Skellam-type spatial processes, and offers tools for modeling complex, non-stationary spatial point patterns with multi-rate, multi-component structure. Overall, the paper broadens the toolkit for spatial stochastic modeling with rigorous probabilistic and analytic characterizations of generalized, time-changed Poisson-type processes.

Abstract

Spatial Poisson point processes on finite-dimensional Euclidean space provide fundamental mathematical tools for modeling random spatial point patterns. In this paper, we introduce and analyze several Poisson-type spatial point processes. In particular, we propose and study a point process, namely, the generalized Poisson random field (GPRF), in which more than one point can be observed with positive probability, within a rectangular region having infinitesimal Lebesgue measure. A thinning of the GPRF into independent GPRFs with reduced rate parameters is discussed. Furthermore, we consider these processes indexed by the positive quadrant of the plane and analyze their fractional variants. Various distributional properties of these processes and related governing differential equations are obtained. Later, we define and analyze a spatial Skellam-type point process via GPRF. Moreover, a fractional variant of it in the two parameter case is studied in detail.

Some Spatial Point Processes of Poisson Family

TL;DR

This paper advances the theory of spatial Poisson-type processes by introducing the generalized Poisson random field (GPRF) on and establishing its representation as a weighted sum of independent Poisson random fields, along with a complete infinitesimal characterization, capacity functional, and thinning results. It extends GPRF to the positive quadrant as a two-parameter Lévy process with rectangular increments, develops integrals and fractional variants via inverse stable subordination, and derives governing fractional differential equations and Wright-function based expressions for the FGPRF. Building on these foundations, the work defines a generalized Skellam point process (GSPP) from independent GPRFs and studies its fractional counterpart, including a compound Poisson field representation and explicit finite-dimensional distributions. The framework unifies 2D time-changed Poisson models, provides exact and fractional dynamics for Skellam-type spatial processes, and offers tools for modeling complex, non-stationary spatial point patterns with multi-rate, multi-component structure. Overall, the paper broadens the toolkit for spatial stochastic modeling with rigorous probabilistic and analytic characterizations of generalized, time-changed Poisson-type processes.

Abstract

Spatial Poisson point processes on finite-dimensional Euclidean space provide fundamental mathematical tools for modeling random spatial point patterns. In this paper, we introduce and analyze several Poisson-type spatial point processes. In particular, we propose and study a point process, namely, the generalized Poisson random field (GPRF), in which more than one point can be observed with positive probability, within a rectangular region having infinitesimal Lebesgue measure. A thinning of the GPRF into independent GPRFs with reduced rate parameters is discussed. Furthermore, we consider these processes indexed by the positive quadrant of the plane and analyze their fractional variants. Various distributional properties of these processes and related governing differential equations are obtained. Later, we define and analyze a spatial Skellam-type point process via GPRF. Moreover, a fractional variant of it in the two parameter case is studied in detail.
Paper Structure (11 sections, 7 theorems, 101 equations)

This paper contains 11 sections, 7 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Generalized Poisson random field
  3. Thinning of a GPRF
  4. GPRF on $\mathbb{R}^2_+$
  5. Thinning of GPRF on $\mathbb{R}^2_+$
  6. Integrals of GPRF on $\mathbb{R}^2_+$
  7. Fractional GPRF on $\mathbb{R}^2_+$
  8. Generalized Skellam point processes
  9. Generalized Skellam point process on $\mathbb{R}^2_+$
  10. Fractional GSPP on $\mathbb{R}^2_+$
  11. Acknowledgement

Key Result

Proposition 2.1

Let $\{N(A),\ A\in\mathcal{A}_d\}$ be a PRF with rate parameter $\lambda>0$, and let $\{S_N(A),\ A\in\mathcal{A}_d\}$ be as defined in (thndef1). Then, $\{S_N(A),\ A\in\mathcal{A}_d\}$ and $\{N(A)-S_N(A),\ A\in\mathcal{A}_d\}$ are PRFs with rate parameters $\lambda p$ and $\lambda(1-p)$, respectivel

Theorems & Definitions (18)

  • Definition 2.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • proof
  • ...and 8 more