Concentration Inequalities for Sub-Weibull Random Tensors

Paper: arXiv 2509.03439

Authors: Yunfan Zhao

Abstract

We extend the theory of concentration inequalities to simple random tensors with heavy-tailed coefficients. Specifically, we consider the class of sub-Weibull distributions $\mathcal{S}_α$ for $α\in [1, 2]$. We establish concentration bounds for Euclidean functions of such tensors, exhibiting a phase transition between sub-gaussian and heavy-tailed regimes. Our results rely on a new Generalized Maximal Inequality for products of heavy-tailed random variables and a martingale analysis using Nagaev-type inequalities.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical extension of tensor concentration inequalities to sub-Weibull (heavy-tailed) distributions using advanced probability techniques like Nagaev-type inequalities and martingale analysis. However, it is entirely theoretical with no empirical backtests, financial data validation, or code provided, resulting in a low empirical rigor score.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Concentration Inequalities for Sub-Weibull Random Tensors α ∈ [1, 2]"] --> B["Inputs:<br>Tensors with Heavy-Tailed/Sub-Weibull Coefficients & Euclidean Functions"]
    
    subgraph Methodology & Analysis
        B --> C["Step 1: Develop Generalized Maximal Inequality for Heavy-Tailed Products"]
        C --> D["Step 2: Apply Martingale Analysis via Nagaev-type Inequalities"]
    end
    
    Methodology & Analysis --> E["Computational Process:<br>Derive Tail Bounds & Analyze Regime Shifts"]
    
    E --> F["Key Outcome:<br>New Concentration Bounds Exhibiting Sub-Gaussian to Heavy-Tailed Phase Transition"]