Skellam Random Field

• Skellam random field is a signed count-valued field defined as the difference of two independent Poisson random fields with stationary, independent rectangular increments.
• It extends to generalized and fractional constructions, allowing richer jump structures and applications in spatial statistics and differential privacy.
• The field is used to model phenomena such as photon count differences in astrophysics and noise in privacy mechanisms, providing precise probabilistic error control.

A Skellam random field is a signed count-valued random field whose values, marginals, or increments are generated by differences of Poisson-type components. In the narrowest and most explicit two-parameter Lévy-field sense, it is the difference of two independent Poisson random fields on $\mathbb{R}_+^2$, $S(s,t)=N_1(s,t)-N_2(s,t)$, so that each $S(s,t)$ has a Skellam law and the field has independent, stationary rectangular increments. In a broader measure-theoretic sense, it is naturally modeled as a random signed measure or signed point process whose Jordan decomposition consists of two independent Poisson or Poisson-type random measures; this yields Skellam-distributed evaluations on bounded sets. The terminology is not fully uniform across the literature: some papers reserve it for the plane Lévy field, whereas others treat the Skellam point process as the canonical signed analogue of the Poisson point process and view a “Skellam random field” as a signed-measure construction built from that object (Vishwakarma, 13 Sep 2025, Passeggeri, 2024, Vishwakarma, 23 Jan 2026).

1. Definition and conceptual scope

The scalar Skellam distribution is the law of a difference of two independent Poisson variables. If $U\sim \mathrm{Poisson}(\lambda_1)$, $V\sim \mathrm{Poisson}(\lambda_2)$, and $W=U-V$, then

$$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$

with $I_k$ the modified Bessel function of the first kind. A Skellam random field lifts this difference-of-Poissons structure from a single random variable to a set-indexed or multi-parameter object.

Two principal usages appear in the cited literature. The first is the direct field construction $S(B)=N_1(B)-N_2(B)$ for measurable sets $B$, with $S(s,t)=N_1(s,t)-N_2(s,t)$0 and $S(s,t)=N_1(s,t)-N_2(s,t)$1 independent Poisson random fields or point processes. The second is a generalized signed field in which the positive and negative parts may themselves be generalized Poisson random fields or completely random measures, so the field remains signed and independently scattered but the jump structure is richer than $S(s,t)=N_1(s,t)-N_2(s,t)$2. In both cases the defining feature is not merely integer-valuedness, but decomposition into independent positive and negative counting components (Vishwakarma, 13 Sep 2025, Vishwakarma, 23 Jan 2026, Passeggeri, 2024).

A common misconception is that “Skellam random field” denotes a single universally standardized object. The current literature does not support that claim. Rather, there is a narrow plane-valued Lévy-field construction, a broader signed-point-process construction, and several generalized or fractional extensions. Another common misconception is that any signed count field with Skellam-looking marginals is a genuine Skellam field. Global constraints, conditioning, or fractional time changes can destroy the simplest independent-increment interpretation, even when Skellam-type formulas remain visible (Sinyukov, 2018, Tathe et al., 11 Apr 2025).

2. Two-parameter Lévy field on $S(s,t)=N_1(s,t)-N_2(s,t)$3

The most explicit field-level definition is the Skellam random field on the positive quadrant. Starting from independent Poisson random fields $S(s,t)=N_1(s,t)-N_2(s,t)$4 and $S(s,t)=N_1(s,t)-N_2(s,t)$5 with rates $S(s,t)=N_1(s,t)-N_2(s,t)$6, one sets

$S(s,t)=N_1(s,t)-N_2(s,t)$7

Here $S(s,t)=N_1(s,t)-N_2(s,t)$8, and the field has rectangular increments

$S(s,t)=N_1(s,t)-N_2(s,t)$9

inherited from the independent Poisson fields. In this formulation the Skellam random field is a two-parameter Lévy field with independent and stationary rectangular increments (Vishwakarma, 13 Sep 2025).

For $S(s,t)$0, its one-point distribution is

$S(s,t)$1

Its first two moments and covariance are

$S(s,t)$2

$S(s,t)$3

The probability generating function is

$S(s,t)$4

The point probabilities satisfy a first-order PDE system; for example,

$S(s,t)$5

This package of independent rectangular increments, explicit Bessel-function marginals, and Lévy-type evolution is the clearest field-theoretic realization of the term (Vishwakarma, 13 Sep 2025).

The same paper introduces a generalized Skellam random field by allowing a finite jump set $S(s,t)$6 and defining

$S(s,t)$7

with independent Poisson random fields $S(s,t)$8. The ordinary Skellam field is the special case $S(s,t)$9. A weak convergence theorem shows that such generalized fields arise as limits of sparse lattice random fields with rare jumps, providing a field-level Poisson limit mechanism rather than only a formal construction (Vishwakarma, 13 Sep 2025).

3. Random signed measures and the Skellam point process

A more foundational treatment replaces field coordinates by measurable-set evaluations. On a localized Borel space $U\sim \mathrm{Poisson}(\lambda_1)$0, a random signed measure is a measurable map

$U\sim \mathrm{Poisson}(\lambda_1)$1

where $U\sim \mathrm{Poisson}(\lambda_1)$2 denotes the space of locally finite countably additive real-valued set functions. Existence of such objects is characterized by three conditions on a countable generating ring: finite additivity on disjoint sets, continuity at the empty set, and local bounded variation. This existence theorem gives a canonical definition of random signed measures and yields uniqueness of the extension (Passeggeri, 2024).

Every random signed measure admits a Jordan decomposition

$U\sim \mathrm{Poisson}(\lambda_1)$3

where $U\sim \mathrm{Poisson}(\lambda_1)$4 and $U\sim \mathrm{Poisson}(\lambda_1)$5 are unique random measures and $U\sim \mathrm{Poisson}(\lambda_1)$6 is the total variation. A completely random signed measure (CRSM) is then defined by independent increments: for disjoint $U\sim \mathrm{Poisson}(\lambda_1)$7, the random variables $U\sim \mathrm{Poisson}(\lambda_1)$8 are jointly independent. The corresponding Kingman-type representation expresses such a field as the sum of fixed atoms, a deterministic signed measure, and a Poisson integral over $U\sim \mathrm{Poisson}(\lambda_1)$9 (Passeggeri, 2024).

Within this framework, the Skellam point process is the canonical signed analogue of the Poisson point process. It is defined as a signed point process whose Jordan decomposition are two independent Poisson point processes; equivalently, it is the difference of two independent Poisson point processes. Consequently, for each $V\sim \mathrm{Poisson}(\lambda_2)$0, $V\sim \mathrm{Poisson}(\lambda_2)$1 is Skellam distributed. The same paper proves the structural equivalence that any simple signed point process with independent increments is a Skellam point process, and conversely any Skellam point process is a simple signed point process with independent increments. This supplies a rigorous measure-theoretic basis for treating Skellam random fields as signed independently scattered random measures rather than only as coordinate-indexed stochastic processes (Passeggeri, 2024).

4. Generalized and fractional constructions

A substantial generalization replaces the underlying Poisson random field by a generalized Poisson random field (GPRF), which allows more than one point to occur with positive probability in an infinitesimal rectangle. With rates $V\sim \mathrm{Poisson}(\lambda_2)$2, the field $V\sim \mathrm{Poisson}(\lambda_2)$3 has probability generating function

$V\sim \mathrm{Poisson}(\lambda_2)$4

mean

$V\sim \mathrm{Poisson}(\lambda_2)$5

variance

$V\sim \mathrm{Poisson}(\lambda_2)$6

and representation

$V\sim \mathrm{Poisson}(\lambda_2)$7

for independent Poisson random fields $V\sim \mathrm{Poisson}(\lambda_2)$8. A generalized Skellam point process is then formed by signed combinations

$V\sim \mathrm{Poisson}(\lambda_2)$9

with independent GPRFs $W=U-V$0. The classical Skellam-type spatial field is the special case $W=U-V$1, namely

$W=U-V$2

with

$W=U-V$3

Its pgf is

$W=U-V$4

Fractionalization enters by inverse-stable time changes. One family defines

$W=U-V$5

another uses one-sided time change $W=U-V$6, and a third takes the difference of two independent fractional Poisson random fields. These models retain explicit point probabilities, governing fractional differential equations, and closed-form first and second moments. For example,

$W=U-V$7

while

$W=U-V$8

The ordinary field is recovered at $W=U-V$9 (Vishwakarma, 13 Sep 2025).

A parallel fractional construction time-changes the generalized Skellam point process by inverse stable subordinators: $$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$0 Its pgf is written with a generalized Wright function,

$$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$1

where

$$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$2

The same paper derives the corresponding mean, variance, covariance, and state-probability formulas. These constructions show that “fractional Skellam random field” is not a single model but a family of subordinated signed fields with nonclassical evolution equations (Vishwakarma, 23 Jan 2026).

A further distinctive result is that the rectangle integral

$$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$3

has a scaled compound Poisson field representation. This identifies the integrated Skellam field as another Poisson-driven signed random field rather than an ad hoc aggregation (Vishwakarma, 13 Sep 2025).

5. Constraints, conditioning, and nonclassical limits

The simplest Skellam field picture assumes independent Poisson components without global constraints. That assumption fails in semi-open systems subject to exact conservation laws. If a total charge-like quantity

$$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$4

is fixed, then subsystem differences $$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$5 are coupled and the subsystem law is not the ordinary Skellam law. A modified Skellam distribution is introduced: $$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$6 with effective parameters $$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$7 and shift $$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$8 chosen to encode the conservation-law coupling. The result suppresses fluctuations relative to the unconstrained case, reduces to ordinary Skellam when the subsystem is tiny, and collapses to a delta distribution when the subsystem becomes the full system (Sinyukov, 2018).

Conditioning can also convert Skellam-type dependence into Gaussian asymptotics. For independent Poisson variables $$P(W=k)=e^{-(\lambda_1+\lambda_2)}\left(\frac{\lambda_1}{\lambda_2}\right)^{k/2}I_k\!\left(2\sqrt{\lambda_1\lambda_2}\right),\qquad k\in\mathbb Z,$$9 with linearly growing parameters, the conditional variable

$I_k$0

has asymptotically Gaussian centered and scaled law: $I_k$1 The proof proceeds through a constrained generating function, Stirling asymptotics, a multidimensional Laplace method, and Hwang’s quasi-powers theorem. This result is directly relevant to Skellam-field settings in which equality constraints couple neighboring or matched differences (Durand et al., 2021).

A different nonclassical regime appears in shuffle privacy. In the critical scaling $I_k$2 and proportional composition $I_k$3, the centered released count

$I_k$4

converges to a Skellam variable $I_k$5 with independent

$I_k$6

The resulting limit experiment is $I_k$7, a Skellam-shift limit with full support on $I_k$8 for interior compositions. Boundary compositions reduce to Poisson-shift limits. This shows that Skellam structure can emerge as a rare-event asymptotic field of two opposing error streams rather than only from a direct Poisson-difference model (Shvets, 10 Mar 2026).

6. Applications and related stochastic models

In high-energy and astrophysical photon counting, Skellam structure arises whenever background subtraction is performed at the count level. For Crab pulsar X-ray photometry, the pulsed main-pulse and interpulse counts are defined by subtracting off-pulse from on-pulse counts. The interpulse histogram is very well fit by a Skellam distribution, with reduced chi-square $I_k$9 for 98 degrees of freedom versus $S(B)=N_1(B)-N_2(B)$0 for a Gaussian fit, and AIC/BIC strongly favor Skellam. The main pulse is also better fit by Skellam than Gaussian, but exhibits a high-count tail and excess variance; two-pulse sums return to Skellam consistency and the lag-1 autocorrelation $S(B)=N_1(B)-N_2(B)$1 yields no physically meaningful memory correlation. This is a direct empirical demonstration that differenced photon counts are naturally modeled by Skellam laws rather than Gaussian surrogates (Worchel et al., 6 Apr 2026).

In differential privacy, the multi-dimensional Skellam mechanism treats a vector of independent Skellam coordinates as a discrete noise field on $S(B)=N_1(B)-N_2(B)$2. For integer-valued queries, the mechanism adds $S(B)=N_1(B)-N_2(B)$3 coordinatewise, satisfies

$S(B)=N_1(B)-N_2(B)$4

and has Rényi differential privacy leading term matching the Gaussian mechanism,

$S(B)=N_1(B)-N_2(B)$5

Its practical advantages are closure under summation and Poisson-based sampling, which make it suitable for secure aggregation and communication-constrained federated learning. This is not a spatial field in the geometric sense, but it is a genuine high-dimensional Skellam-valued random field indexed by coordinates (Agarwal et al., 2021).

Approximation theory and dependent-process theory broaden the context. Differences of weakly dependent, Poisson-like counts in networks and image processing admit explicit Skellam approximation bounds via Stein’s method, including settings with spillover and dependence between the two counts (Gan et al., 2017). Sums of symmetric Markov-dependent three-point variables can be approximated by Skellam compound Poisson laws in local, total variation, and Wasserstein metrics (Čekanavičius et al., 2024). Temporal analogues include Skellam-GARCH processes, in which the conditional law is Skellam and the latent second-moment recursion yields a unique stationary regime together with geometrically decaying $S(B)=N_1(B)-N_2(B)$6-mixing coefficients under contractive conditions (Doukhan et al., 2020). One-dimensional fractional and order-$S(B)=N_1(B)-N_2(B)$7 process extensions, such as the Skellam process of order $S(B)=N_1(B)-N_2(B)$8, its space-fractional and tempered variants, and the fractional Skellam process of order $S(B)=N_1(B)-N_2(B)$9, supply a rich process-level toolkit from which spatial or space-time Skellam fields can be built by replacing a one-parameter clock with a multi-parameter or set-indexed random measure (Gupta et al., 2020, Kataria et al., 2021).

Across these constructions, the persistent structural idea is the same: a Skellam random field is a signed stochastic field built from two independent positive counting mechanisms, with the classical plane Lévy field, the signed-point-process formalism, and generalized or fractional Poisson-type variants forming a hierarchy rather than competing definitions.