Central limit theorem for $ε$-independent products and higher-order tensors

TL;DR

The paper develops a central limit theorem for sums of products of $\epsilon$-independent random variables, encoding the evolving dependence via graphon limits. It introduces a general framework using $\mathcal{M}_L$-decorated graphons to capture mixtures of independence structures and derives a master CLT for grid-graph indices, with explicit moment formulas expressed through decorated intersection graphs. The results recover classical and free CLTs (L=1) and extend to higher-order tensor products, including a tensor-free CLT as a special case, all within a unified, graphon-based approach. This framework broadens the scope of CLTs in noncommutative probability and connects graphon theory with mixed independence, enabling concise derivations for complex dependency topologies.

Abstract

We establish a central limit theorem (CLT) for families of products of $ε$-independent random variables. We utilize graphon limits to encode the evolution of independence and characterize the limiting distribution. Our framework subsumes a wide class of dependency structures and includes, as a special case, a CLT for higher-order tensor products of free random variables. Our results extend earlier findings and recover as a special case a recent tensor-free CLT, which was obtained through the development of a tensor analogue of free probability. In contrast, our approach is more direct and provides a unified and concise derivation of a more general CLT via graphon convergence.

Figures (2)

• Figure 1: Example of lexicographical product.
• Figure 2: Example of $\pi$ and $\ker(\theta)$.

Theorems & Definitions (23)

• Theorem 1.1: CSY
• Theorem 1.2: lancien2024centrallimittheoremtensor
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Lemma 2.1: speicherjanusz2016mixture
• Lemma 2.2
• proof
• Theorem 3.1
• Corollary 3.2