Universal Robust Limit Theorem

• Universal robust limit theorem is a framework that establishes convergence in distribution using sublinear expectation to handle model uncertainty.
• It extends classical limit theorems by utilizing nonlinear Lévy processes and PIDEs to capture both jump and diffusion behaviors.
• Probabilistic approximation schemes with explicit error bounds enable robust inference in finance, insurance, and applied mathematics.

A universal robust limit theorem is a mathematical result that establishes the convergence in distribution of normalized sums (or other functionals) of random variables or high-dimensional objects to a universal limiting law, under a framework that allows for model or distributional uncertainty. Such theorems generalize classical probabilistic limit results by accommodating nonlinear expectations, ambiguity in probability laws, and complex uncertainty sets, thereby offering a unified approach to robust inference in probability, statistics, combinatorics, and applied mathematics.

1. Foundations in Sublinear Expectation and Model Uncertainty

A central innovation in universal robust limit theorems is the adoption of sublinear expectation frameworks, most notably Peng’s G-expectation. Unlike classical expectation, which is linear and based on a single probability measure, sublinear expectation operates over a collection of (potentially mutually singular) probability measures. This nonlinearity allows for the modeling of epistemic uncertainty without committing to a single law, embedding monotonicity, constant preservation, sub-additivity, and positive homogeneity as core properties. Distributions are encoded as functionals:

$$F^X[\varphi] = \hat{\mathbb{E}}[\varphi(X)], \qquad \varphi \in \mathrm{Cb, Lip}(\mathbb{R}^n),$$

where $\hat{\mathbb{E}}$ denotes the sublinear expectation, and the law of $X$ is specified only by its action on bounded, Lipschitz functionals. This framework is pivotal in developing robust central limit theorems and α-stable laws that are valid under both moment and structural uncertainty, as in (2205.00203, 2301.07819, 2310.02134), and (2506.18374).

2. Nonlinear Lévy Processes and Robust Limit Laws

The limiting objects in many universal robust limit theorems are nonlinear Lévy processes, characterized not by a single Lévy triplet but by an uncertainty set $\Theta$ of triplets $(F_\mu, q, Q)$, corresponding to jump, drift, and diffusion parameters, respectively. For $\alpha \in (1,2)$, prototypical random sequences

$$\left\{ \left( \frac{1}{\sqrt{n}}\sum_{i=1}^n X_i,\, \frac{1}{n}\sum_{i=1}^n Y_i,\, \frac{1}{n^{1/\alpha}}\sum_{i=1}^n Z_i \right) \right\}_{n=1}^\infty$$

converge in distribution (under sublinear expectation) to a process $$\tilde{L}_t = (\tilde{\xi}_t, \tilde{\eta}_t, \tilde{\zeta}_t)_{t \in [0,1]}$$, where each component is a nonlinear robust analog of the classical counterparts (2205.00203, 2506.18374). For $\alpha \leq 1$, as in (2301.07819), non-integrable (even heavy-tailed) cases are included.

These processes are characterized by fully nonlinear (and possibly degenerate) partial integro-differential equations (PIDEs) of the form:

$$\left\{ \begin{array}{l} \displaystyle \partial_t u(t,x,y,z) - \sup_{(F_\mu, q, Q) \in \Theta} \Bigg\{ \int_{\mathbb{R}^d} \delta_\lambda u(t,x,y,z)\, F_\mu(d\lambda) + \langle D_y u, q \rangle + \frac{1}{2} \mathrm{tr}[D^2_x u\, Q] \Bigg\} = 0, \ u(0,x,y,z) = \phi(x,y,z), \end{array} \right.$$

where the nonlocal increment operator for jumps is

$$\delta_\lambda u(t,x,y,z) := u(t,x,y,z+\lambda) - u(t,x,y,z) - \langle D_z u(t,x,y,z), \lambda \rangle.$$

This equation captures the entire uncertainty set, allowing for robust, nonlinear modeling of both continuous and jump behavior (2205.00203, 2506.18374).

3. Weak Convergence and Lévy–Khintchine Representation

The establishment of convergence relies on a weak compactness approach within the sublinear expectation framework. The key analytic step is to demonstrate tightness and weak convergence for the sequence of normalized sums or increments, even under non-integrability (as with $\alpha \in (0,1]$ in (2301.07819)). Technical tools include mollification, truncation, consistency estimates, and estimation of “δ-moment” conditions that replace classical integrability.

A significant byproduct is a generalized Lévy–Khintchine representation. In the robust, nonlinear setting, the characteristic exponent is replaced by a sublinear generator:

$$\mathcal{G}(u)(t,x) = \sup_{F_\mu \in \mathcal{L}} \int_{\mathbb{R}^d} \delta_{\lambda}^{\alpha} u(t,x) F_\mu(d\lambda),$$

where $\mathcal{L}$ is the set of uncertain α-stable Lévy measures. The jump term

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

allows for precise description of limiting laws and connects the probabilistic and PDE formulations robustly (2301.07819).

4. Probabilistic Approximation, Numerical Methods, and Error Analysis

A major practical advance is the construction of probabilistic approximation schemes—recursive, piecewise-constant methods that approximate the viscosity solution of the limiting PIDE. For a time-step $h > 0$, a standard scheme is

$$u_h(t, x, y, z) = \begin{cases} \phi(x, y, z), & t \in [0, h), \ \hat{\mathbb{E}}\left[ u_h(t-h, x + h^{1/2} X, y + h Y, z + h^{1/\alpha} Z) \right], & t \in [h,1], \end{cases}$$

where $(X, Y, Z)$ has the same law as the single-step increment under sublinear expectation (2506.18374).

These schemes allow for explicit, quantitative error bounds, leading to Berry–Esseen-type results of the form

$$-\mathcal{C}_1 n^{-1/2} - \min_{\varepsilon>0}(4\mathcal{C}_1 \varepsilon + E_1(\varepsilon, 1/n)) \leq \hat{\mathbb{E}}\left[ \varphi\left(\frac{S_n^1}{\sqrt{n}}, \frac{S_n^2}{n}, \frac{S_n^3}{n^{1/\alpha}}\right) \right] - \tilde{\mathbb{E}}\left[\varphi(\xi_1, \eta_1, \zeta_1)\right] \leq \mathcal{C}_2 n^{-1/2} + \min_{\varepsilon>0}(4\mathcal{C}_2 \varepsilon + E_2(\varepsilon, 1/n)),$$

where $S_n^i$ are partial sums, $E_{1,2}$ are explicit functions of discretization and mollification parameters, and $\varphi$ is a test function (2506.18374, 2310.02134). The convergence rates and bounds subsume classical CLT, robust CLT (Peng), and robust α-stable limit theorems as special cases.

5. Unification of Robust Limit Theorems

The universal robust limit theorem framework, as detailed in (2506.18374) and related works, unifies Peng's robust CLT (for diffusion uncertainty), classical laws of large numbers, and α-stable robust limit theorems (for jump and heavy-tailed uncertainty). At the methodological core is the nonlinear PIDE with an uncertainty set that may feature non-separable, coupled jump-diffusion uncertainty, thereby covering new regimes inaccessible to traditional numerical or probabilistic results.

This unification allows simultaneous quantification of convergence rates across all three classic regimes (Gaussian, law of large numbers, and heavy-tailed α-stable laws) under model uncertainty and establishes explicit error bounds for each.

6. Practical and Theoretical Implications

• Numerical Implementation: The fully probabilistic and recursive nature of the approximation schemes makes them well-suited for implementation in a variety of computational settings. The explicit error bounds offer guarantees for the accuracy of the numerical solution and limit law approximation under uncertainty (2310.02134, 2506.18374).
• Robust Inference: By eschewing reliance on a single underlying law, the results provide firm theoretical grounding for robust inference under distributional ambiguity—a critical concern in finance, insurance, risk management, and applied probability.
• Modeling Heavy Tails: The relaxation of integrability conditions and the use of α-stable Lévy measures (with $\alpha \le 2$) accommodate regimes with heavy-tailed or non-integrable data, expanding the range of robust limits (2301.07819).
• Unified View of Uncertainty: Treating both jump and diffusion uncertainties in a coupled, non-separable fashion, the universal robust limit theorem offers a more general and practically relevant approach than methodologies that address only classical, Gaussian or independent regimes.

7. Summary Table: Key Features Across Robust Limit Results

Paper/Setting	Framework	Limiting Law	PIDE Characterization?	Error/Rate Quantified?
(2205.00203)	Sublinear expectation	Nonlinear Lévy process	Yes	Yes
(2506.18374)	G-expectation, α-stable	Nonlinear Lévy process	Yes	Yes (explicit)
(2310.02134)	Discrete approx., robust	α-stable process	Yes (viscosity sol.)	Yes (quantitative)
(2301.07819)	No integrability needed	Nonlinear α-stable	Yes	Yes (general)

These results collectively demonstrate a comprehensive, quantitative, and numerically accessible theory for robust limit theorems across a broad spectrum of uncertainty-affected probabilistic models. The universal robust limit theorem thus stands as a unifying principle for weak convergence under uncertainty, enabling new advances in both the theory and practice of robust statistical inference.