Stability-Based CLT: Robust α-Stable Laws

• Stability-Based CLT is a framework that uses system stability properties to establish central limit results in heavy-tailed and uncertain environments.
• It employs sublinear expectation and viscosity solutions of nonlinear integro-differential equations to capture nonlinear α-stable behaviors without strict integrability.
• The approach generalizes the classical CLT by integrating model uncertainty through a jump-uncertainty set, facilitating analysis of complex stochastic systems.

A stability-based Central Limit Theorem (CLT) refers to any central-limit or weak-invariance principle whose limit process and convergence mechanism fundamentally exploit the algebraic, analytic, or geometric stability properties of the system under consideration—typically, stability of the limit law (e.g., α-stable distributions), stability of fixed points or attractors in dynamical systems, or stability properties of PDEs or operators associated with the limits. This concept has catalyzed key generalizations of the classical CLT both in abstract probability and in the presence of uncertainty, nonlinearity, or lack of integrability.

1. Foundations: Stability, Sublinear Expectation, and the Robust α-Stable CLT

A robust α-stable CLT under sublinear expectation is formulated in the setting $(\Omega, \mathcal{H}, \mathcal{E})$, where $\mathcal{E}$ is a sublinear expectation. For $\alpha \in (0, 1]$, let $\{Z_i\}_{i=1}^\infty$ be an i.i.d. sequence of $\mathbb{R}^d$-valued random variables under $\mathcal{E}$. The main result asserts that for the α-stable normalization,

$$n^{-1/\alpha} S_n = n^{-1/\alpha} \sum_{i=1}^n Z_i,$$

the sequence converges in law to $\tilde{\zeta}_1$, where $(\tilde{\zeta}_t)_{t \in [0,1]}$ is a multidimensional nonlinear symmetric α-stable Lévy process determined by a jump uncertainty set $\mathcal{L}$. The limit is characterized through a fully nonlinear partial integro-differential equation (PIDE),

$$\partial_t u(t,x) - \sup_{F \in \mathcal{L}} \left\{ \int_{\mathbb{R}^d} \delta_\lambda^\alpha u(t,x) F(d\lambda) \right\} = 0,\quad u(0,x) = \phi(x),$$

where the jump increment operator $\delta_\lambda^\alpha$ takes the form

$$\delta_\lambda^\alpha u(t,x)= \begin{cases} u(t,x+\lambda)-u(t,x), & \alpha \in (0,1) \ u(t,x+\lambda)-u(t,x) - \langle D_x u(t,x), \lambda\rangle \mathbf{1}_{\{|\lambda|\leq 1\}}, & \alpha=1 \end{cases}$$

The key technical achievement is the absence of strict integrability conditions on $\{Z_i\}$, with tightness established under a relaxed $\varepsilon$-moment control and a model-consistency condition for the small-jump expansion (Jiang et al., 2023).

2. Core Components: Sublinear Expectation, Independence, and Model Uncertainty

The sublinear expectation structure satisfies monotonicity, constant preservation, subadditivity, and positive homogeneity. Independence is defined analogously to the classical expectation: $Y$ is independent of $X$ if

$$\mathcal{E}[f(X,Y)] = \mathcal{E}[\mathcal{E}[f(x,Y)]_{x=X}]$$

for all bounded measurable $f$. Identically distributed sequences and independence enable the construction and analysis of normalized sums in the broad context of model uncertainty. The jump-uncertainty set $\mathcal{L}$ collects all α-stable Lévy measures of the form

$$F(d\lambda) = \int_{S^{d-1}} |r|^{-1-\alpha} dr\, \mu(d\theta),~~ \lambda = r \theta,$$

where $\mu$ varies over a convex compact set of measures on the sphere, enforcing uniform upper and lower bounds for the measure $\mu(S^{d-1})$. This set models ambiguity in both radial jump intensity and jump direction (Jiang et al., 2023).

3. PIDE Characterization and Lévy–Khintchine Representation

The nonlinear symmetric α-stable process $\tilde{\zeta}_t$ is characterized by the nonlinear generator

$$\mathcal{G}[\phi](x) = \sup_{F \in \mathcal{L}} \int \delta_\lambda^\alpha \phi(x) F(d\lambda)$$

acting on smooth test functions $\phi$. The process’ law is dictated by this generator, paralleling the classical Lévy–Khintchine formula. For test functions with $\phi(0)=0$,

$$\lim_{t\to 0} t^{-1} \mathbb{E}[\phi(\tilde{\zeta}_t)] = \sup_{F\in\mathcal{L}} \int \delta_\lambda^\alpha \phi(0) F(d\lambda).$$

This construction is robust to heavy tails, using truncation schemes for controlling the contribution of rare large jumps and demonstrating weak convergence of normalized partial sums even in the absence of finite first moments (Jiang et al., 2023).

4. Truncation, Tightness, and the Weak Convergence Method

The development employs a truncation device: for truncation threshold $N$, define $Z^N_i = Z_i \mathbf{1}_{|Z_i| \leq N}$. Tightness is first proved for the sequence $\{n^{-1/\alpha} S_n\}$ via an $\epsilon$-moment bound (i.e., finiteness of $\sup_{n} \mathcal{E}[|n^{-1/\alpha} S_n|^\epsilon]$ for some $\epsilon < \alpha$). A Donsker-type procedure constructs the limiting process on Skorokhod space, with finite-dimensional distributions governed by $\mathcal{G}$. Consistency of the truncated processes and their error bounds yield convergence to the viscosity solution of the governing PIDE, establishing both existence of the nonlinear process and its characterization (Jiang et al., 2023).

5. Stability Identity and the Emergence of Nonlinear α-Stable Laws

The stability property under sublinear expectation is preserved: for independent copies $X \cong Y$,

$$a X + b Y \cong (a^\alpha + b^\alpha)^{1/\alpha} X,\quad a,b \geq 0,$$

mirroring the defining property of classical strictly stable laws. Scaling of the nonlinear process satisfies

$$\tilde{\zeta}_t \cong t^{1/\alpha} \tilde{\zeta}_1,$$

and symmetry $\tilde{\zeta}_1 \cong -\tilde{\zeta}_1$ is inherited directly from the jump symmetry of elements in $\mathcal{L}$. The process incorporates model uncertainty by enveloping all possible α-stable jump intensities within specified bounds, producing a family of nonlinear limit laws parametrized by $\mathcal{L}$ (Jiang et al., 2023).

6. Reduction to the Classical CLT and Comparison

If the model uncertainty collapses — i.e., $\mathcal{L} = \{F_0\}$ is a singleton — the PIDE reduces to its linear counterpart,

$$\partial_t u - \int \delta_\lambda^\alpha u(t,x) F_0(d\lambda) = 0,$$

so that $\tilde{\zeta}_1$ is a classical linear α-stable random variable. Thus, the stability-based sublinear CLT strictly generalizes the classical Lindeberg–Feller theorem for heavy-tailed summands, and recovers it in the case of no model ambiguity (Jiang et al., 2023).

7. Technical Synthesis and Methodological Significance

Stability-based CLTs in the robust, heavy-tailed regime uniquely exploit the combination of nonlinear expectation, sensitivity to model uncertainty via $\mathcal{L}$, and a PDE/integral-operator toolkit. The framework sidesteps explicit computation of characteristic functions, accommodating non-integrable data by instead working through viscosity solutions and limiting behaviors of empirical processes. The stability mechanism serves both as a probabilistic regularizer and as the source of the nontrivial scaling limit, tightly linking the theory of nonlinear Lévy processes and fully nonlinear integro-differential equations (Jiang et al., 2023).

8. Connections, Extensions, and Generalizations

The analysis aligns with the broader trend in probability, stochastic analysis, and PDE theory to handle uncertainty and non-uniqueness at the process level. Related approaches have been extended to settings of mean-uncertainty (G-expectations and G-Brownian motion), multidimensional robust martingale central limit theorems, and stable limit laws for processes and functionals in random environments (Guo et al., 2023, Rokhlin, 2015). The stability-based technique is a cornerstone for the rigorous analysis of aggregation, limiting distributions, and uncertainties in stochastic systems characterized by non-integrability, heavy tails, and model ambiguity.

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Reference:

"A robust $\alpha$-stable central limit theorem under sublinear expectation without integrability condition" (Jiang et al., 2023)