Mixed Poisson-Stable Distributions

• The mixed Poisson-Stable family is a class of discrete probability distributions constructed by mixing a Poisson process with stable or heavy-tailed mixing variables, enabling flexible modeling of count data.
• It unifies traditional models like the negative binomial and Poisson-inverse Gaussian by leveraging common probabilistic structures and analytic techniques, including compound Poisson representations.
• The framework supports applications in duality analysis for coagulation-fragmentation processes, limit theorems, and complex modeling in fields such as finance, biology, and Bayesian nonparametrics.

The mixed Poisson-stable family comprises a broad class of discrete probability laws constructed by mixing the Poisson distribution with a stable or stable-like mixing distribution. This family arises by considering the Poisson parameter (the mean) as a random variable—often with heavy-tailed, even real-valued, or stable mixing distributions—and includes many classical and generalized models for overdispersed, heavy-tailed, or multifractal count data. The family admits strong structural, analytic, and probabilistic unification of discrete stable distributions, partition-valued processes, species sampling models, composition schemes in combinatorics, fractional processes, multidimensional mixed regimes, and novel compound Poisson representations. Its canonical members, such as the broadly discrete stable distributions, inherit properties intrinsic to stable laws—such as domain of attraction, heavy tail, self-similarity, infinite divisibility—while conforming to the algebraic and stochastic rules of counting processes and discrete thinning.

1. Definition, Construction, and Canonical Examples

The defining construction of the mixed Poisson-stable family is as follows. Given a random variable $X$, possibly stable or heavy-tailed, and a (potentially large) scale parameter $\rho>0$, let

$Y \mid X \sim \mathrm{Poisson}(\rho X)$

with the understanding that $X$ may have support on $\mathbb{R}$ (provided integrability and tail constraints are satisfied). Marginalizing over $X$ yields the mixed Poisson law: $$P(Y = k) = \int \frac{(\rho x)^k}{k!} e^{-\rho x} dF_X(x), \qquad k \geq 0,$$ with $F_X$ the distribution function of $X$.

A canonical case sets $X$ as an extreme stable random variable $ES(\alpha,\sigma,\delta)$ (maximal right skew, index $0<\alpha\leq2$). The probability generating function (PGF) is

$$G(z) = \exp\left\{ (z-1)\delta - \operatorname{sec}\left(\frac{\pi\alpha}{2}\right)\sigma^\alpha (1-z)^\alpha \right\}$$

for $\alpha\neq 1$, or

$$G(z) = \exp\left\{(z-1)\delta + \frac{2\sigma}{\pi}(1-z)\log(1-z)\right\}$$

for $\alpha=1$ (Townes, 5 Sep 2025, Townes, 24 Jul 2024).

A central constraint is that $\delta \geq \alpha \gamma$ (with $\gamma$ the negative coefficient of the $(1-z)^\alpha$ term) to ensure the function is a valid PGF (Townes, 24 Jul 2024, Townes, 5 Sep 2025).

The family includes as special or limiting cases:

• Negative binomial (gamma mixing)
• Poisson-inverse Gaussian (inverse Gaussian mixing)
• Rayleigh and Mittag-Leffler mixed Poisson models
• Broadly discrete stable distributions (when $X$ is stable with maximal skewness).

2. Discrete Stable Laws: Generalization and Structural Identities

Classically, discrete analogs of strictly stable laws were restricted to $0<\alpha\leq 1$, using binomial thinning as the analog of scaling. The mixed Poisson-stable family provides a generalization that holds for all $0<\alpha\leq2$, replacing thinning with a Poisson translation and allowing for both heavy- and light-tailed regimes (Townes, 5 Sep 2025).

The discrete stability property is characterized by PGFs of the form

$$G(z) = \exp\left\{(z-1)\delta + \gamma(1-z)^\alpha \right\}$$

under the constraint $\delta \geq \alpha \gamma$. These are discretely infinitely divisible (DID) and admit compound Poisson representations: $G(z) = \exp\left\{ \lambda (H(z) - 1) \right\}$ where $H(z)$ is the PGF of a "broad Sibuya" distribution: $$H(z) = \begin{cases} 1 - [\rho(1-z) + (1-\rho)(1-z)^\alpha] & \text{if } \alpha\neq1 \ z + \rho(1-z)\log(1-z) & \text{if } \alpha=1 \end{cases}$$ with parameter regimes for $\rho$ depending on $\alpha$ (Townes, 5 Sep 2025). For $\rho=0$, this recovers the classical Sibuya summands; for $\rho\neq0$, the support is generalized to the full range $\alpha\in(0,2]$.

Discrete self-decomposability holds if $\delta \geq \alpha^2\gamma$ for $\alpha\neq1$ (or $\delta\geq 2\gamma$ for $\alpha=1$), implying unimodality (Townes, 5 Sep 2025).

3. Coagulation-Fragmentation Dualities and Partition-Valued Processes

A haLLMark of the continuous stable world is the multiplicative duality between fragmentation and coagulation processes, epitomized by the classic Pitman two-parameter Poisson–Dirichlet (PD$(\alpha,\theta)$) duality. The mixed Poisson-stable family supports a rich generalization of these dualities:

• Via the mixed Poisson–Kingman class, partitions of mass generated by jumps of stable subordinators (and their Poisson-tilted mixtures) admit explicit, bridge-based coagulation and fragmentation operators (James, 2010).
• The key identity, for mixing variable $\zeta$ and bridges $Q_{*,*}$, is

$$Q_{\alpha, \tau_\delta(\zeta)} \circ Q_{\delta, \zeta} \equiv Q_{\alpha\delta, \zeta}$$

• The duality diagram for mass partitions is:

$$\mathcal{P}_\delta(\zeta) \xrightarrow{\mathrm{Coag}} \mathcal{P}_\alpha(\tau_\delta(\zeta)) \xleftrightarrow{\mathrm{Frag}} \mathrm{PD}(\alpha,-\alpha\delta)$$

• In the most general extensions, duality is realized for any subordinator, and multi-group structured partitions are constructed via the Poisson Hierarchical Indian Buffet Process (PHIBP), admitting explicit compound Poisson representations for all four components: fine partition, its coagulation, time-forward fragmentation, and time-backward coalescence (James, 26 Aug 2025).

The family underpins the theory of Gibbs partitions (Ho et al., 2018), Poisson-Dirichlet bridges, and partially exchangeable partition structures used in Bayesian nonparametrics, species sampling, and population genetics.

4. Limit Theorems, Moment Structure, and Analytic Combinatorics

Mixed Poisson-stable laws are characterized by explicit relations between moment sequences:

• If $Y$ has a mixed Poisson law with mixing variable $X$ and scale parameter $\rho$:

$$\mathbb{E}[(Y)_{\underline{s}}] = \rho^s \mathbb{E}[X^s]$$

and the moment generating functions are related by

$$\varphi(z) = \psi(\rho(e^z-1)), \quad \psi(z) = \varphi(\log(1+z/\rho))$$

(Kuba et al., 2014).

• As $\rho\to\infty$, $Y/\rho$ converges in moments and distribution to $X$; after centering and scaling, $(Y/\rho - X)\sqrt{\rho}$ converges in law to a normal variance mixture $\mathcal{N}(0,X)$ (Kuba, 12 Feb 2025, Barreto-Souza et al., 2020).
• In combinatorics, core size parameters and counts of substructures exhibit mixed Poisson convergence, often with mixing laws expressible in terms of generalized Mittag-Leffler, Rayleigh, or stable distributions, stemming from the composition of singular generating functions. Notably, phase transitions occur: the (properly normalized) counts either converge to continuous heavy-tailed limits (when scaling diverges) or to mixed Poisson-stable laws (when scaling is finite) (Banderier et al., 2021).

5. Fractional, Multivariate, and Mixed Regimes

Fractional discrete processes—such as time-changed compound Poisson processes using stable subordinators or their inverses—exhibit overdispersion and heavy tails. Their transition probabilities solve fractional Kolmogorov equations: $$D_t^\nu p_k(t) = -A p_k(t) + A \sum_j q_j p_{k-j}(t)$$ with $D_t^\nu$ a Caputo or Riemann–Liouville fractional derivative; such processes belong to the mixed Poisson-stable family by virtue of their compound/subordinated structure (Beghin et al., 2013).

Multivariate constructions, including correlated multivariate (mixed) Poisson processes, are naturally included: the backward simulation framework allows for arbitrary marginal mixed Poisson laws and a prescribed correlation structure, with efficient generation of sample paths tractable via convex combinations of extreme joint distributions (Chiu et al., 2020).

Limit theorems in stochastic geometry (e.g., in random geometric graphs) demonstrate "mixed" convergence: some statistics converge to Poisson limits, others to Gaussian, yielding joint mixed Poisson–stable regimes (Bourguin et al., 2012).

6. Beyond Nonnegative Mixing: Real-Valued and Broad Extensions

The common constraint of nonnegative mixing distributions is not essential. Provided the left tail is light (subexponential) and negative mass is small, real-valued mixing variables (including stable laws with support $\mathbb{R}$ for $\alpha>1$) yield valid mixed Poisson distributions, as long as certain tail and normalizing conditions (e.g., $\delta$ above a critical threshold) are met (Townes, 24 Jul 2024). This extension broadens the modeling scope, allowing, for example, the Poisson–extreme stable family to capture both positive and negative jumps and extreme heavy tails inaccessible with nonnegative mixing (Townes, 5 Sep 2025, Townes, 24 Jul 2024).

The "broadly discrete stable" class includes all tail indices $0<\alpha\leq2$, is discretely infinitely divisible, and its members are discretely self-decomposable and unimodal under explicit parameter constraints (Townes, 5 Sep 2025).

7. Applications and Theoretical Impact

The mixed Poisson-stable family serves as a unifying class for modeling:

• Overdispersed or heavy-tailed count data (finance, insurance, biology, quantitative linguistics) (Townes, 24 Jul 2024, Barreto-Souza et al., 2020)
• Statistical properties of random combinatorial structures (increasing trees, recursive maps, lattice path statistics) (Kuba et al., 2014, Banderier et al., 2021)
• Partition-valued and feature allocation models, with applications to Bayesian nonparametrics, topic models, and microbiome analysis (James, 2010, James, 26 Aug 2025)
• Long-range dependence and multifractality in stochastic processes (mixed fractional Poisson, subordinated point processes) (Kataria et al., 2019)
• General limit theory for random sums, with scale and regime transitions described via stable or exponential family mixtures (Barreto-Souza et al., 2020, Kuba, 12 Feb 2025).

These models provide rigorous convergence results, analytical tractability for moment and cumulant computation, closed-form inference techniques (method of moments, EM-based maximum likelihood), and compound Poisson representations through novel generalizations of the Sibuya law (Townes, 5 Sep 2025).

The theoretical framework supports dualities (fragmentation/coagulation), product-form partition probability functions, and the extension of classical stable domains of attraction to the discrete and mixed regime—offering new directions for probabilistic modeling, inference, and simulation in both applied and theoretical domains.