Discrete Stable Distributions

• Discrete stable distributions are integer-valued probability laws that generalize continuous stable laws using techniques like binomial thinning and Poisson translation.
• They incorporate compound Poisson representations, algebraic normalization, and pseudo-summation frameworks to extend stability across a broad range of tail indices.
• Applications in econometrics, finance, scientometrics, and coding theory offer practical methods for modeling count data with both heavy and light tails.

Discrete stable distributions constitute a class of probability laws that generalize strict stability from the real line to the discrete setting, enabling the rigorous modeling of count-valued data with both heavy and light tails. Much of their mathematical structure is inherited from analogues of stable distributions, but with modifications—such as binomial thinning, Poisson translation, and pseudo-summation—necessary to accommodate discrete domains. Recent advances extend the classical (Steutel–van Harn) version to include broader parameter ranges, compound Poisson representations with generalized Sibuya summands, and connections to algebraic stability concepts and random summation schemes. This article surveys foundational aspects, generalizations, algebraic frameworks, domains of attraction, compound representations, and applications, referencing key results from (Townes, 5 Sep 2025, Klebanov et al., 2010, Klebanov et al., 2014, Slámová et al., 2015, Alexeev et al., 2023), and related literature.

1. Classical and Generalized Definitions

Classically, discrete stability is defined for nonnegative integer-valued random variables via thinning operations that mimic scalar multiplication. For random variable $X$, binomial thinning replaces scaling, i.e., $$X \mapsto a \circ X = \sum_{i=1}^X \varepsilon_i(a)$$ (where $\varepsilon_i(a)$ are i.i.d. Bernoulli$(a)$), and the distribution $X$ is (strictly) discrete stable if for every $n$ there exists $p(n)$ such that

$X \stackrel{d}{=} X_1(p) + X_2(p) + \dots + X_n(p)$

with independent copies. The probability generating function (PGF) satisfies

$P(z) = P((1-p) + pz)^n$

leading to the explicit form

$$P(z) = \exp[-\lambda (1-z)^\gamma], \quad \lambda > 0, \quad \gamma \in (0,1]$$

This restriction to $\gamma \leq 1$ (or the strict stable index $\alpha \in (0,1]$) reflects preservation of integer values (Klebanov et al., 2014).

Recently, (Townes, 5 Sep 2025) generalizes the definition to all tail indices $\alpha \in (0,2]$ by supplementing thinning with Poisson translation and constructing mixed Poisson-stable families. This "broadly discrete stable" class uses PGFs of the form

$G(z) = \exp[(z-1)\delta + \gamma (1 - z)^\alpha]$

and is exactly the law of a mixed Poisson variable with extreme stable mixing ([Poi ∧ 𝒮(α, σ, δ)]).

Other generalizations substitute thinning/portlying operators (Slámová et al., 2015), or normalize via associative Look-Up Tables to define pseudo-sumps (Alexeev et al., 2023), or via random (ν-stable) summation indices (Klebanov et al., 2010).

2. Compound Poisson Representations and Broad Sibuya Distributions

Discrete infinite divisibility (DID) implies that all discrete stable laws can be represented as compound Poisson distributions. Specifically, (Townes, 5 Sep 2025) provides

$G(z) = \exp[\lambda (H(z) - 1)]$

where $H(z)$ is the PGF of a summand law—termed the broad Sibuya (bSib) distribution. For $\alpha \neq 1$,

$$H(z) = 1 - [\rho(1 - z) + (1 - \rho)(1 - z)^\alpha]$$

and for $\alpha = 1$,

$H(z) = z + \rho (1 - z) \log(1 - z)$

with suitable $\rho$ ensuring validity as a PGF. The classical Sibuya distribution appears as a special case for $\gamma \leq 1$.

Self-decomposability and unimodality of discrete stable distributions require extra constraints on parameters; for example, $δ ≥ α^2γ$ for $\alpha \ne 1$ guarantees unimodal, self-decomposable laws (see (Townes, 5 Sep 2025), Prop. "Discrete self-decomposability").

3. Algebraic and Pseudo-Summation Frameworks

Alternative approaches to discrete stability employ algebraic normalization or generalized summation. "Casual stability" (Klebanov et al., 2014, Klebanov, 2015), uses random normalization (via sequences of PGFs or Laplace transforms), leading to stability conditions such as

$L(s) = L^n(-\log[g_n(s)])$

Algebraic normalization can also be characterized via characteristic functions: a distribution $f(t)$ is $\gamma$-casually strictly stable if for some normalizing sequence $g_n(t)$,

$f(t) = g^n(-i \log[g_n(t)])$

There exists a limit theorem showing this form approximates discrete stable limits.

Pseudo-summation operations, as in (Alexeev et al., 2023), use associative Look-Up Tables to define a general binary operation $\oplus$; stable laws in this context must satisfy self-similarity under the operation:

$$\xi_1 \oplus \xi_2 \stackrel{d}{=} b \oplus \xi + a$$

The domains of attraction and the types of infinitely divisible distributions are characterized for any associative operation, including classical sum, maximum, and modulo operations.

4. ν-Stable Distributions and Stability Under Random Summation

The ν-stable concept (Klebanov et al., 2010) extends strict stability by randomizing the number of summands. For $X$, define a family $\{\nu_p\}$ with $\mathbb{E} \nu_p = 1/p$, and require

$$X \stackrel{d}{=} p^{1/\alpha} \sum_{i=1}^{\nu_p} X_i$$

Here, stability depends on commutativity of the semigroup generated by $\mathcal{P}_p(z) = \mathbb{E}[z^{\nu_p}]$. The solution is obtained via analytic iteration methods; when $\mathcal{P}_p$ is related to Chebyshev polynomials, the characteristic function is

$f(t) = 1/\cosh(a t)$

yielding the hyperbolic secant distribution for strictly ν-normal case. The approach admits further generalizations to geometric summation, branching schemes, and random recursive models.

5. Tempered Discrete Stable Distributions

To address overly heavy tails of classical discrete stable models, several tempered variants have been introduced (Barabesi et al., 2016, Grabchak, 2017). The tempered discrete Linnik and Poisson–Tweedie classes use altered Lévy measure or PGF to dampen the tail, e.g.,

$$g_X(s) = [1 + sgn(a) b d ((1 - c s)^a - (1 - c)^a)]^{-1/d}$$

The DTS class (Grabchak, 2017) is defined by subordinating a Poisson process to a tempered stable variable $T \sim PTS_\alpha(q, n)$:

$$E[s^{N_T}] = \exp\left(-n \int_0^\infty (1 - e^{-(1-s)x}) q(x) x^{-1-\alpha} dx \right)$$

These distributions offer finite moments and domain-of-attraction results for sums of iid heavy-tailed (but ultimately tempered) variables.

6. Applications and Modeling Significance

Discrete stable and tempered discrete stable distributions are increasingly used for count data in econometrics, finance, scientometrics (e.g., citation counts), environmental statistics, population genetics, and risk management. The broad discrete stable family provides flexible models accommodating both the heavy-tailed center and temperate extremes of empirical distributions. Random summation, thinning-based autoregressive processes, and compound Poisson–stable constructions (with Sibuya or broad Sibuya summands) admit practical interpretability and simulation methods.

For instance, in citation modeling (Klebanov et al., 2014), composition of geometric and Sibuya processes produces discrete stable laws that fit empirical data, explaining discrepancies between means and medians. Mixed Poisson–stable laws (Townes, 5 Sep 2025) extend the modeling toolkit beyond classical absolute continuous frameworks, allowing both light and heavy tails, unimodal or multimodal count distributions, and explicit compound representations. In coding theory, pseudo-summation limit laws (Alexeev et al., 2023) provide rigorous foundations for quantized iterative algorithms.

7. Summary

Discrete stable distributions encompass a spectrum of models for integer-valued data, generalizing strict stability to discrete domains via thinning, Poisson translation, compound Poisson formulations, and algebraic operations. Extensions to full tail index ranges, new summand distributions (broad Sibuya), and algebraic or pseudo-summation frameworks have expanded both theoretical understanding and practical modeling capabilities. Recent results confirm that domains of attraction, infinite divisibility, unimodality, and self-decomposability are precise and flexible under these generalized settings. Applications in econometrics, coding theory, scientific rating, and beyond have established discrete stable laws as a vital part of modern probabilistic modeling of count data.