Iterative Randomized Lindeberg Method

• The Iterative Randomized Lindeberg Method is a framework that replaces complex components with simplified ones to obtain quantitative bounds on the convergence to limit laws.
• It integrates techniques from Stein's method and Fourier analysis with randomized iterative algorithms to ensure robust error control in high-dimensional settings.
• The method finds practical applications in numerical optimization, network design, and establishing explicit rates in central limit and stable law approximations.

The Iterative Randomized Lindeberg Method is a conceptual framework and set of algorithmic techniques that combine the classical Lindeberg principle from probability theory with iterative randomized algorithmic paradigms. The unifying theme is to approach problems of limit theorems—such as quantitative central limit approximations and stable law convergence—by iteratively and often randomly replacing components of a complex structure (typically a sum of random variables or vectors) with simpler or idealized components, while carefully measuring and accumulating the error. This approach has enabled both highly quantitative bounds in high-dimensional probability and robust, scalable random algorithms in numerical computation and optimization.

1. Classical Lindeberg Principle and Quantitative Extensions

The classical Lindeberg principle provides a replacement strategy to prove central limit theorems (CLTs) by successively swapping variables in a sum with Gaussian variables and showing that the global distribution approaches the normal law under the Lindeberg condition. The quantitative extension, such as in "Stein's method and a quantitative Lindeberg CLT for the Fourier transforms of random vectors" (Berckmoes et al., 2013), refines this dichotomous (yes–no) convergence with explicit, numerical bounds on the deviation from normality.

A key quantitative tool is the Lindeberg index:

$$\text{Lin}(\{E_{n,k}\}) = \sup_{\epsilon>0} \limsup_{n\to\infty} \sum_{k=1}^n \mathbb{E}[E_{n,k}^2; |E_{n,k}| > \epsilon]$$

Explicit inequalities such as

$\Delta(H_n \to E) \le 2\,\text{Lin}(\{E_{n,k}\})$

where $\Delta$ measures maximal discrepancy in Fourier transforms and Lin quantifies deviation from Lindeberg's condition, form the analytic backbone for iterative error control.

In an approach-theoretic context, these bounds are dimension-robust and facilitate error-tracking in infinite-dimensional or high-dimensional limit setting.

2. Stein's Method and Iterative Error Estimates

Stein's method, which characterizes the normal distribution as the solution to a differential equation (the Stein equation), allows one to measure the difference between the law of a sum and the Gaussian law. For multivariate random vectors, adopting the Fourier test function $e_t(x) = \exp(-i \langle t, x \rangle)$ yields an integral representation for the Hessian of the solution:

$$\text{Hess}\ f_{e_t}(x) = -\frac{1}{2}\, t t^\top \int_0^1 \exp\{-i\sqrt{s}\langle t, x \rangle - \tfrac{1}{2}(1-s)|t|^2\}\ ds$$

This representation avoids singularities inherent in the classical approach (removing problematic factors such as $1/(1-s)$), allowing precise control over error terms.

By bounding terms involving the Hessian, one can iteratively estimate the rate at which the distribution of partial sums approaches Gaussianity, thereby enabling systematic refinement or "randomization" at each step based on the residual deviation measured via the index $\text{Lin}(\{E_{n,k}\})$.

3. Iterative Randomized Algorithmic Frameworks

The iterative randomized paradigm finds analogous structure in randomized algorithms for linear systems, as seen in "Randomized Iterative Methods for Linear Systems" (Gower et al., 2015). There, a general iterative update for solutions to $A x = b$ is performed by randomly "sketching" or selecting constraints, then projecting the current solution to satisfy these randomly chosen constraints while minimizing deviation in a problem-dependent geometry:

$$x^{k+1} = x^k - B^{-1}A^\top S (S^\top AB^{-1}A^\top S)^\dagger S^\top (A x^k - b)$$

where $B$ defines the geometry and $S$ is randomly sampled.

The convergence rate is governed by spectral quantities, e.g.,

$$\rho = 1 - \lambda_{\text{min}}(B^{-1/2} \mathbb{E}[Z] B^{-1/2})$$

and is exponential for typical cases. By judiciously choosing or randomizing the sketching matrix $S$, and tuning the geometry $B$, one may design efficient iterative algorithms with rigorous rates—analogous to replacing components in the Lindeberg principle.

A plausible implication is that by tracking the cumulative approximation error via an index analogous to Lin$(\{E_{n,k}\})$, one could dynamically adjust the randomness or the direction of updates, yielding a robust, adaptive iterative randomized Lindeberg method for both probabilistic and numerical settings.

4. Extensions to Stable Laws and Non-Gaussian Limits

Traditional CLT methods based on Fourier analysis struggle with stable laws, particularly when moments fail to exist (e.g., for $\alpha \in (0,1]$). "Approximation to the stable law by Lindeberg principle" (Chen et al., 2018) replaces characteristic function technology with a tailored "Taylor-like expansion" and an analysis based on the Kolmogorov forward equation:

$$\mathbb{E}[f(Z + aX)] - \mathbb{E}[f(Z)] - \mathbb{E}[aX]\,\mathbb{E}[f'(Z)] = -a^\alpha\,\mathbb{E}[\mathcal{A}_{\alpha,\beta} f(Z)] + R(a)$$

where $\mathcal{A}_{\alpha,\beta}$ is a nonlocal operator modeling the infinitesimal generator of the stable process.

In the iterative randomized Lindeberg spirit, one replaces each summand individually and controls the accumulated approximation error, often in smooth Wasserstein distances. This iterative swapping is a randomized process, delivering explicit rates and error bounds without reliance on Fourier analysis—extending the principle to heavy-tailed, non-Gaussian limit distributions.

5. Metrics, Bounds, and Nonstandard Settings

The iterative randomized Lindeberg method is characterized by its ability to deliver explicit bounds in various probability metrics. As shown in "Approximate central limit theorems" (Berckmoes et al., 2016), approximate CLTs resulting from small but nonzero Lindeberg index values provide bounds in Kolmogorov, Wasserstein, and parametrized Prokhorov distances:

$$d(\sum E_{n,k}, N(0,1)) \le C\cdot \text{Lin}(\{E_{n,k}\})$$

with $C=1$ for Kolmogorov (under Feller's condition), $C=8$ for Wasserstein, and $C=4$ for Prokhorov. These bounds quantify not only the rate but also the magnitude of deviation from the normal law in nonstandard settings, where classical CLT statements are otherwise silent.

The approach is inherently iterative—local errors from each step or summand replacement are summed, and the process is randomized in that expectation, rather than pointwise control, governs error bounds. This facilitates extension to complex, high-dimensional, and triangular array settings.

6. Algorithmic Manifestations and Network Design Connections

Randomized iterative Lindeberg-style methods are not confined to pure probability; their algorithmic structure appears in combinatorial optimization—most notably in network design via iterative randomized rounding (Angelidakis et al., 2021). In these applications, an LP relaxation is iteratively randomized, with partial solutions sampled and contracted in rounds, and progress towards optimal connectivity is tracked through deterministic invariants (such as witness trees). Error or approximation quality is tightly bounded at each round via a randomized process guided by expectations and witness structure, yielding improved guarantees (e.g., a $1.892$-approximation for node-connectivity augmentation) and transparent analyses.

A plausible implication is that the Lindeberg principle's iterative replacement and error measurement scheme may serve as an underlying blueprint for constructing and analyzing randomized rounding and iterative algorithms in broad settings, including high-dimensional optimization and robust network design.

7. Broader Implications and Future Directions

The iterative randomized Lindeberg method, as developed through quantitative CLT extensions, randomized iterative algorithms, and robust combinatorial optimization, exemplifies a general strategy: replacing, at each iteration, complex components by tractable or idealized ones; measuring and controlling error via index-based or spectral bounds; and iterating the process until convergence. This yields explicit rates, dimension-robust bounds, and a unified framework for handling non-classical limits and high-dimensional problems.

The method opens the possibility for further generalization to infinite-dimensional settings, adaptive algorithms where error indices dynamically adjust the replacement rule, and connections to numerical and combinatorial randomized algorithms where probabilistic replacement and error tracking are central. Its capacity to rigorously and adaptively quantify convergence makes it a foundational tool for both probability theory and randomized computation.