Skellam Point Process Overview

• Skellam point process is a signed counting process defined as the difference between two independent Poisson processes, providing a framework for modeling bidirectional jumps.
• It extends to fractional dynamics, order-k constructions, and non-homogeneous settings, enabling analysis of anomalous diffusions and neural spike train resets.
• The process admits a compound Poisson representation with detailed Lévy measures, facilitating studies on long-range dependence and statistical inference in various applications.

Searching arXiv for recent and foundational papers on Skellam point processes and related variants. A Skellam point process is a signed counting process built from differences of Poisson counting mechanisms. In its classical form, it is defined by

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

where $N_1$ and $N_2$ are independent homogeneous Poisson processes with intensities $\lambda_1,\lambda_2>0$. At fixed time $t$, $S(t)$ has the Skellam distribution, while as a process it is a pure-jump Lévy process with upward and downward jumps corresponding to the two Poisson drivers. Subsequent work has extended this construction to order-$k$ and arbitrary-jump models, non-homogeneous and multivariate settings, inverse-subordinator and stable-subordinator time changes, spatial and multiparameter random fields, and biologically motivated reset mechanisms for neural spike trains (Buchak et al., 2018, Cinque et al., 10 Apr 2025, Ramezan et al., 2022).

1. Classical construction and basic probabilistic structure

The classical Skellam process is the difference of two independent Poisson processes,

$S(t)=N_1(t)-N_2(t), \qquad t\ge 0.$

Its one-dimensional law is

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

where $I_k$ is the modified Bessel function of the first kind. The process has mean and variance determined by the two Poisson rates, and its infinitesimal behavior is that a jump of $N_1$0 occurs at rate $N_1$1 and a jump of $N_1$2 occurs at rate $N_1$3 (Buchak et al., 2018).

This process is also a compound Poisson process. One representation is

$N_1$4

where $N_1$5 with

$N_1$6

That representation makes explicit that the Skellam process is a signed counting process rather than merely a static distribution on $N_1$7 (Lee et al., 2022).

Its state probabilities satisfy the forward system

$N_1$8

with $N_1$9 and $N_2$0 for $N_2$1. The corresponding Lévy measure is concentrated at $N_2$2, so the classical model is the two-sided unit-jump member of a much wider family of signed jump processes (Gupta et al., 2020).

2. Time changes, fractional dynamics, and nonlocal evolution

A major line of development replaces calendar time by a random operational time. For an inverse subordinator

$N_2$3

the time-changed process

$N_2$4

has marginal probabilities

$N_2$5

where $N_2$6 is the density of the inverse subordinator. These marginals satisfy the nonlocal system

$N_2$7

with the generalized Caputo-Djrbashian convolution derivative $N_2$8. When $N_2$9, this reduces to the known fractional Skellam equation (Buchak et al., 2018).

The literature distinguishes two fractional Skellam constructions. In type 1, one takes the difference of two independent fractional Poisson processes,

$\lambda_1,\lambda_2>0$0

In type 2, one time-changes the Skellam process itself,

$\lambda_1,\lambda_2>0$1

with $\lambda_1,\lambda_2>0$2 the inverse stable subordinator. The two models have different moderate deviation behavior, and the paper on noncentral moderate deviations emphasizes that type 2 is “less random” in a sense because there is only one inverse-stable randomization instead of two (Lee et al., 2022).

Further generalization leads to the generalized fractional Skellam process

$\lambda_1,\lambda_2>0$3

where $\lambda_1,\lambda_2>0$4 is a generalized Skellam process and $\lambda_1,\lambda_2>0$5 is an independent inverse stable subordinator. Its state probabilities satisfy a Caputo-fractional system, its probability generating function has Mittag-Leffler form, and the one-dimensional distributions are not infinitely divisible. The same paper derives long-range dependence for the generalized fractional Skellam process and short-range dependence for its increment process (Kataria et al., 2021).

For order-$\lambda_1,\lambda_2>0$6 models, the fractional Skellam process of order $\lambda_1,\lambda_2>0$7,

$\lambda_1,\lambda_2>0$8

inherits grouped jump sizes $\lambda_1,\lambda_2>0$9 while replacing the ordinary time derivative in the state equations by a Caputo derivative. Its correlations decay as $t$0, which establishes long-range dependence (Kataria et al., 2021).

3. Generalized, order-$t$1, and non-homogeneous families

A broad generalization replaces the two jump sizes $t$2 by an arbitrary fixed jump alphabet. One formulation is

$t$3

where $t$4 and the $t$5 are independent non-homogeneous Poisson processes with cumulative rates

$t$6

The probability generating function is

$t$7

This family is closed under scaling and summation, admits law-of-large-numbers and central-limit-type limits, and in the homogeneous case has a compound Poisson representation with Lévy measure concentrated on the jump sizes $t$8 (Cinque et al., 10 Apr 2025).

The order-$t$9 construction specializes this idea to symmetric positive and negative jumps up to size $S(t)$0. The Skellam process of order $S(t)$1 is

$S(t)$2

where $S(t)$3 and $S(t)$4 are independent Poisson processes of order $S(t)$5. Its marginal law retains the Skellam-Bessel form but with time scale multiplied by $S(t)$6, and its Lévy measure is supported on $S(t)$7. This model generalizes the classical Skellam process by allowing jump magnitudes beyond one (Gupta et al., 2020).

A different generalization uses generalized counting processes with jump sizes $S(t)$8 and possibly time-dependent rates. The non-homogeneous generalized Skellam