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arxiv: 2509.10870 · v2 · submitted 2025-09-13 · 🧮 math.PR

Skellam Random Fields and Their Fractional Variants

Pradeep Vishwakarma This is my paper

Pith reviewed 2026-05-18 16:48 UTC · model grok-4.3

classification 🧮 math.PR
keywords Skellam random fieldLévy processRiemann integralcompound Poisson fieldfractional variantsweak convergencecharacteristic functionspatial point process
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The pith

The Riemann integral of a Skellam random field equals a scaled compound Poisson field whose characteristic function has an explicit form.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a Skellam random field as a two-parameter Lévy process with rectangular increments over the positive quadrant of the plane and establishes its weak convergence. It then analyzes the Riemann-Liouville integral of this field over finite rectangles and shows that the integral admits a scaled compound Poisson field representation. An explicit expression for the characteristic function of the integral is derived. The study further examines three fractional variants of the field and derives their point probabilities along with associated governing equations and distributional properties.

Core claim

A Skellam random field on the positive quadrant is introduced as a two-parameter Lévy process with rectangular increments. Its weak convergence is obtained, and the Riemann-Liouville integral over finite rectangles is characterized as a scaled compound Poisson field. An explicit formula for the characteristic function of this integral is given, and three fractional variants are analyzed for their point probabilities, governing equations, and other distributional features.

What carries the argument

The Skellam random field, defined as a two-parameter Lévy process with rectangular increments on the positive quadrant, which supports the weak convergence result and enables the scaled compound Poisson characterization of its Riemann integral.

If this is right

  • The Riemann integral of the SRF over finite rectangles admits a scaled compound Poisson field representation.
  • The characteristic function of this integral can be expressed in closed form for direct evaluation.
  • Three fractional variants of the SRF possess explicit point probabilities and satisfy specific governing equations.
  • Further distributional properties of the fractional variants follow from the base SRF analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The compound Poisson representation may allow efficient generation of sample paths for the integrals without direct integration.
  • The rectangular increment structure could extend naturally to modeling other spatial count processes with Lévy-type dependence.
  • The fractional variants might link to time-fractional or space-fractional point process models already used in statistics.

Load-bearing premise

The Skellam random field must behave as a two-parameter Lévy process with rectangular increments on the positive quadrant for the weak convergence and the subsequent integral analysis to hold.

What would settle it

A direct calculation or simulation for a small rectangle where the distribution or characteristic function of the Riemann integral of the SRF fails to match the predicted scaled compound Poisson field.

Figures

Figures reproduced from arXiv: 2509.10870 by Pradeep Vishwakarma.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original abstract

We study some Skellam-type spatial point processes. As a particular case, we consider a Skellam random field (SRF) on the positive quadrant of the plane, which is a two parameter L\'evy process with rectangular increments. A weak convergence result is obtained for the SRF. The Riemann-Liouville integral of the SRF over finite rectangles is analyzed. We derive a scaled compound Poisson field characterization for the Riemann integral of the SRF. Also, an explicit expression of its characteristic function is obtained. Later, we consider three fractional variants of the two parameter SRF. Their point probabilities, associated governing equations, and various other distributional properties are analyzed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper defines a Skellam random field (SRF) as a two-parameter Lévy process on the positive quadrant with independent rectangular increments that are Skellam distributed (difference of independent Poisson random measures). It establishes a weak convergence result for the SRF, analyzes the Riemann-Liouville integral over finite rectangles to obtain a scaled compound Poisson field characterization together with an explicit characteristic function, and studies three fractional variants of the SRF, deriving their point probabilities, governing equations, and additional distributional properties.

Significance. If the derivations hold, the work provides a concrete extension of multiparameter Lévy process theory to the Skellam setting and its fractional counterparts. The scaled compound Poisson representation and characteristic function for the integrated field supply explicit analytic tools, while the governing equations for the fractional cases offer a basis for further study in spatial stochastic processes.

minor comments (3)
  1. [Abstract] The abstract states that three fractional variants are considered but does not name or briefly describe them; adding one sentence identifying the specific fractional operators (e.g., Riemann-Liouville, Caputo, or other) would improve readability.
  2. [Definition of SRF] In the section introducing the SRF, the precise parameter values or intensity measures for the underlying Poisson random measures should be stated explicitly when the Skellam increment distribution is first defined, to make the subsequent Lévy-Khintchine application fully traceable.
  3. [Weak convergence] The weak convergence result is announced without an accompanying statement of the topology or the limiting object; a short remark clarifying whether convergence is in finite-dimensional distributions or in a Skorokhod-type space on rectangles would help readers assess the strength of the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript on Skellam random fields and their fractional variants, as well as for the positive assessment of its significance in extending multiparameter Lévy process theory. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs the Skellam random field directly from the definition of a two-parameter Lévy process with independent rectangular Skellam increments on the positive quadrant. All subsequent results—the weak convergence, the scaled compound Poisson characterization of the Riemann-Liouville integral, the explicit characteristic function via Lévy-Khintchine, and the distributional properties of the three fractional variants—follow by applying standard properties of Lévy sheets and fractional integrals to this definition. No equation reduces to a fitted parameter renamed as a prediction, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled in; the derivations remain self-contained against external stochastic-process theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard assumptions from the theory of Levy processes and fractional integrals, with the SRF introduced as a defined object rather than an empirically fitted one.

axioms (2)
  • domain assumption The Skellam random field is a two-parameter Levy process with independent rectangular increments.
    Invoked in the definition of the SRF to obtain weak convergence and integral properties.
  • standard math Standard properties of Riemann-Liouville integrals apply to the SRF over finite rectangles.
    Used to analyze the integral and derive the compound Poisson characterization.
invented entities (1)
  • Skellam random field (SRF) no independent evidence
    purpose: To serve as a spatial point process model with Skellam-type increments in two dimensions.
    Defined as a particular case of Skellam-type spatial point processes in the abstract.

pith-pipeline@v0.9.0 · 5629 in / 1385 out tokens · 47618 ms · 2026-05-18T16:48:24.352704+00:00 · methodology

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Reference graph

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