Discrete Strict Stability

• Discrete strict stability is a property of discrete systems ensuring that operations like thinning or contraction preserve the system's behavior under iteration and perturbation.
• It is foundational in probability theory through thinning operators, in dynamical systems via tangential stability, and in operator algebras by analyzing semiprojectivity in group C*-algebras.
• Applications span robust statistical modeling, discrete-time control systems, and stability analysis in geometric variational problems.

Discrete strict stability refers to a property of discrete systems or distributions whereby a specific operation (such as thinning, substitution, or contraction) ensures that the system retains its essential behaviour under iteration, perturbation, or combination. The concept appears in several mathematical contexts, including probability, dynamical systems, semigroup theory, operator algebras, and geometric analysis. This article presents a comprehensive overview of discrete strict stability, organizing developments in its modern theory, key definitions, analytical frameworks, structural results, and implications for applications.

1. Discrete Strict Stability in Probability Theory

Discrete strict stability for random variables replaces the classical scaling operation with the thinning operator, adapted to integer-valued laws. When $X_1,\ldots,X_n$ are IID copies of a non-negative integer-valued random variable $X$, $X$ is called discrete strictly stable if, for each $n$, there is $a_n \in [0,1]$ such that

$a_n \circ (X_1 + \cdots + X_n) \overset{d}{=} X,$

where $a_n \circ$ denotes the thinning operation: each unit in the sum is independently retained with probability $a_n$ (Aldridge, 15 Sep 2025).

Steutel and Van Harn established that under nondegeneracy, $a_n = n^{-1/\alpha}$ for some $\alpha \in (0,1]$, and that the alternate probability generating function (APGF) of discrete strictly stable laws is

$\psi_X(t) = \exp\left(-\gamma t^\alpha\right)$

for suitable $\gamma$ and $\alpha$ (Aldridge, 15 Sep 2025).

Discrete weakly stable (or simply discrete stable) laws generalize this by replacing the classical shift with a Poisson shift. Specifically, $X$ is stable if, for all $n$, there are $a_n \in [0,1]$ and $b_n \in [0,\infty)$ such that

$$a_n \circ (X_1 + \cdots + X_n) \overset{d}{=} X \oplus b_n,$$

where $X \oplus b_n := X + Z$, $Z$ an independent Poisson($b_n$) variable. This framework encompasses the Poisson and Hermite distributions and leads to the Poisson-delayed Sibuya laws as canonical discrete stable distributions. The APGF in this broader case assumes forms such as

$$\psi_X(t) = \exp\left(-\delta t - \gamma t^\alpha\right)\quad (\alpha \in (0,2]),$$

with certain parameter constraints (Aldridge, 15 Sep 2025).

2. Discrete Strict Stability in Dynamical Systems

In discrete-time dynamical systems governed by monotone convex maps $f:\mathcal{D}\to\mathbb{R}^n$, tangential stability ("t-stability") is defined for a fixed point $v$ by considering the directional derivative $$f'_v(x)=\lim_{\epsilon \to 0^+} [f(v+\epsilon x)-f(v)]/\epsilon$$ (Akian et al., 2010). The point $v$ is t-stable if, for every $x$, the sequence $[(f'_v)^k(x)]_{k\ge 0}$ is bounded above. This is formalized as:

$$\forall x \in \mathbb{R}^n,\; \sup_k (f'_v)^k(x) < +\infty,$$

which is a weaker requirement than Lyapunov stability.

The set of t-stable fixed points forms a convex inf-semilattice under the binary operation

$x \wedge_f y = \lim_{k\to\infty} f^k(x \wedge y),$

where $(x \wedge y)_i = \min\{x_i, y_i\}$, and further structure is described by projection to the critical graph associated with $f$.

Under suitable boundedness hypotheses for the recession map

$$\hat{f}(x) = \lim_{\lambda \to \infty} \lambda^{-1} f(\lambda x),$$

every orbit of $f$ converges to a Lyapunov stable periodic orbit, with periods that are divisors of the cyclicity of the critical graph—effectively matching the order of a permutation of $n$ elements (Akian et al., 2010).

3. Discrete Strict Stability in Operator Algebras and Groups

Discrete strict stability appears in the paper of $C^*$-algebras associated to discrete groups. A group is $C^*$-stable if its full group $C^*$-algebra is semiprojective: every approximate representation can be perturbed to an exact representation (Eilers et al., 2018). Criteria and invariants for stability include winding number obstructions for homogeneous relations and K-theoretic conditions.

Key findings include:

• Finitely generated virtually free groups are $C^*$-stable.
• Virtually abelian groups are $C^*$-stable if and only if the abelian subgroup has rank $\leq 1$.
• The classification of crystallographic groups and various Baumslag-Solitar groups into stable and non-stable cases. Such analysis informs both the stability of representations and the structure of liftings in operator algebras (Eilers et al., 2018).

4. Discrete Strict Stability in Geometry and Variational Methods

In geometric contexts, discrete strict stability arises in the paper of minimal submanifolds and calibrated cones. A minimal cone $C$ is strictly stable if the smallest eigenvalue $d_0(C)$ of the stability operator $L = \Delta^\perp + \mathcal{B}$ is strictly positive:

$d_0(C) = \frac{(n-2)^2}{4} + \lambda_1 > 0,$

where $\lambda_1$ is the first eigenvalue of the Laplacian on the link (Dimler et al., 9 Sep 2024).

For special Lagrangian cones and coassociative cones, strict stability is proved by excluding the existence of marginal Jacobi fields via analytic and spectral methods. In contrast, complex calibrated cones (e.g., zero sets of homogeneous holomorphic polynomials) admit marginal Jacobi fields, and thus are stable but not strictly stable (Dimler et al., 9 Sep 2024).

In discrete differential geometry, variational approaches characterize strict stability for equilibrium planar curves, identifying regular polygons as discrete constant curvature objects and analyzing stability via second variation and discrete Jacobi operators (Jikumaru, 2020).

5. Analytical Frameworks and Methods

A range of methodologies supports discrete strict stability analysis:

• Alternate probability generating functions and functional equations for discrete stable distributions (Aldridge, 15 Sep 2025).
• Directional derivatives and tangential maps to characterize t-stability (Akian et al., 2010).
• Lattice operations (thinning, portlying) for generalizing scaling in discrete settings (Slámová et al., 2015).
• Carleman estimates and CGO solutions in discrete inverse problems for robust uniform stability (Ervedoza et al., 2011).
• Spectral and quadratic Lyapunov function methods in the stability analysis of discrete-time linear complementarity systems (Raghunathan et al., 2020).
• Use of inf-semilattice structures and critical graphs in discrete convex monotone dynamical system analysis (Akian et al., 2010).
• Topological fixed-point arguments and contraction conditions in robust nonlinear discrete-time systems (Zoboli et al., 2022).
• Duality and coherent risk measures in risk-aware stability frameworks for stochastic discrete-time systems (Chapman et al., 2022).

6. Structural Properties, Classification, and Uniqueness

Discrete strict stability often produces rigid algebraic and combinatorial structures:

• The periods of t-stable periodic points correspond to divisors of permutation orders (critical graph cyclicity).
• The set of t-stable fixed points forms a convex inf-semilattice, with uniqueness guaranteed under critical graph conditions (Akian et al., 2010).
• In probability, discrete strictly stable distributions correspond to specific families (Poisson-Sibuya, Hermite) whose domains of normal attraction and representation properties are explicit (Aldridge, 15 Sep 2025, Slámová et al., 2015).
• In operator algebras, group-theoretic properties (virtual freeness, abelian rank, nilpotency) determine $C^*$-stability (Eilers et al., 2018).

7. Applications and Implications

Discrete strict stability has broad and deep implications:

• Provides rigorous underpinnings for robust algorithms and numerical methods (e.g., for inverse problems on discrete grids (Ervedoza et al., 2011)).
• Informs theoretical foundations for representation theory, operator algebra liftings, and soficity in groups (Eilers et al., 2018).
• Enables modeling of heavy-tailed discrete data in statistics, insurance, finance, and network modeling (Aldridge, 15 Sep 2025, Slámová et al., 2015).
• Guides the design of robust discrete-time controllers, especially under sampling and perturbation effects (Park et al., 2019, Zoboli et al., 2022).
• Underlies geometric stability analysis for variational problems and minimal surfaces with discrete structure (Dimler et al., 9 Sep 2024, Jikumaru, 2020).

8. Comparative Perspectives and Extensions

Discrete strict stability is distinct from but related to continuous stability notions, with characteristic modifications (thinning vs scaling, Poisson shifts vs constant shifts). The discrete setting commonly involves:

• Replacement of smooth structures with operations that respect the underlying lattice.
• Extension of classical definitions to accommodate the constraints and idiosyncrasies of discreteness (inf-semilattices, combinatorial objects, alternate PGFs).
• Sensitivity to boundary effects, perturbations, and the combinatorial structure of the underlying set or group.

Recent research continues to extend discrete strict stability theory to multidimensional settings, dependent processes, new classes of operators, generalized stability notions (weak, risk-aware, metric), and analytic/numeric frameworks suited for large-scale and high-dimensional data.

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Discrete strict stability encompasses a mathematically rich and highly structured family of phenomena. It delivers analytic tractability, algebraic rigidity, robustness, and flexibility in modeling discrete systems and distributions, providing theoretical infrastructure for applications from stochastic processes to control theory, operator algebras, and discrete geometry.