A note on the renormalization group approach to the Central Limit Theorem (2303.13905v3)

Abstract: Two proofs of the Central Limit Theorem using a renormalization group approach are presented. The first proof is conducted under a third moment assumption and shows that a suitable renormalization group map is a contraction over the space of probability measures with a third moment. The second proof uses Lyapunov stability and works under a second moment condition. These are by far not the most optimal proofs of the CLT, and the main interest of the proofs are their existence, the CLT being the simplest case in which a renormalization group argument should apply. None of the tools used in this note are new. Similar proofs are known amongst expert in limit theorems, but explicit references are not so easy to come by for non-experts in the field.

• The paper demonstrates that the renormalization group transformation acts as a contraction (constant 2^(-1/2)) on a Fourier-based metric space, ensuring convergence toward a Gaussian distribution.
• It compares two methodologies—the Banach approach with quantifiable contraction properties and the Lyapunov approach that guarantees global asymptotic stability.
• The study outlines implications for extending renormalization techniques to complex probabilistic models, paving the way for future research in advanced convergence frameworks.

An Analysis of the Renormalization Group Approach to the Central Limit Theorem

This paper presents a meticulous paper of the Central Limit Theorem (CLT) through the lens of the renormalization group (RG) approach, with particular emphasis on a third moment assumption. The CLT, a cornerstone of probability theory, traditionally relies on an ensemble of independent and identically distributed (i.i.d.) random variables with finite variance to assert convergence in distribution toward a Gaussian distribution. In contrast, this note expounds on an alternative method using RG, offering insights into its viability for addressing similar probabilistic convergence scenarios.

Methodological Overview

The work meticulously delineates two prevailing methodologies in the application of RG to CLT: the Banach approach and the Lyapunov approach. Each method holds specific theoretical appeal and practical efficacy, influenced by the nuances of measure and metric selections.

1. Banach Approach: This method leverages the contraction principle, utilizing a metric space of probability measures—denoted $Q_3$—equipped with the Fourier-based metric $d_3$. It demonstrates that a transformation $T$, associated with the renormalization map, is a contraction with contraction constant $2^{-1/2}$. This property not only provides clear quantitative results but also sets a foundation for proving convergence of the transformed sequences towards a Gaussian distribution, contingent on the completion of this metric space.
2. Lyapunov Approach: This approach emphasizes the global asymptotic stability of the Gaussian fixed point under the RG transformation. The paper showcases the formulation of a Lyapunov function $V$ that demonstrably decreases under the transformation $T$, ensuring convergence via the notion of decreasing level sets. This approach is more flexible, accommodating scenarios needing only a second moment condition, yet it generally provides less specific quantitative estimates than the Banach method.

Outcomes and Implications

The primary outcome of this exploration is the establishment of the feasible application of the RG approach to CLT under prescribed conditions. Although none of the tools or proofs are substantively new, providing a formal delineation serves as an essential touchstone for both current and future research involving RG in probabilistic contexts.

The implications of employing a renormalization technique to prove widely recognized limit theorems exhibit potential across various domains within mathematics and physics, where many phenomena exhibit convergence behavior akin to the CLT. While not extending significantly beyond existing methodologies, this paper paves the way for enhanced understanding and application of RG in probabilistic limits, particularly within frameworks demanding high moment assumptions.

Prospects for Future Research

The integration of RG in establishing probabilistic limit theorems unveils a gateway for advanced theoretical exploration and potential expansion to more complex probabilistic models, such as those involving dependent data structures or systems manifesting intricate external constraints. Further research could focus on optimizing metric completions and exploring alternative RG transformations that might extend its applicability to broader conditions and weaker assumptions.

In conclusion, the paper presents a rigorous examination of utilizing renormalization group strategies in the context of the CLT, augmenting current literature with precise methodological insights and reinforcing the mathematical underpinning necessary for continued exploration in this intersection of analysis and probability theory.