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A New Proof of Sub-Gaussian Norm Concentration Inequality

Zishun Liu, Sam Power, Yongxin Chen

TL;DR

The paper introduces the averaged moment generating function (AMGF) as a direct tool to study the concentration of sub-Gaussian norms, avoiding the traditional union-bound via an $\varepsilon$-net. By averaging MGFs over the unit sphere, the AMGF serves as a surrogate for $\mathbb{E}(e^{\lambda\|X\|})$ and yields explicit, parameterized concentration bounds for both vectors and matrices through careful Bessel-function analysis. The results improve constants relative to ε-net approaches and generalize to the operator norm of sub-Gaussian matrices without independence assumptions, highlighting the AMGF’s potential for broader probabilistic insights. Overall, the work provides a tighter, more direct methodology for norm concentration with implications for high-dimensional probability and random matrix theory.

Abstract

We present a new method for proving the norm concentration inequality of sub-Gaussian variables. Our proof is based on an averaged version of the moment generating function, termed the averaged moment generating function. Our method applies to both vector cases to bound the vector norm and matrix cases to bound the operator norm. Compared with the widely adopted $\varepsilon$-net technique-based proof of the sub-Gaussian norm concentration inequality, our method does not rely on the union bound and promises a tighter concentration bound.

A New Proof of Sub-Gaussian Norm Concentration Inequality

TL;DR

The paper introduces the averaged moment generating function (AMGF) as a direct tool to study the concentration of sub-Gaussian norms, avoiding the traditional union-bound via an -net. By averaging MGFs over the unit sphere, the AMGF serves as a surrogate for and yields explicit, parameterized concentration bounds for both vectors and matrices through careful Bessel-function analysis. The results improve constants relative to ε-net approaches and generalize to the operator norm of sub-Gaussian matrices without independence assumptions, highlighting the AMGF’s potential for broader probabilistic insights. Overall, the work provides a tighter, more direct methodology for norm concentration with implications for high-dimensional probability and random matrix theory.

Abstract

We present a new method for proving the norm concentration inequality of sub-Gaussian variables. Our proof is based on an averaged version of the moment generating function, termed the averaged moment generating function. Our method applies to both vector cases to bound the vector norm and matrix cases to bound the operator norm. Compared with the widely adopted -net technique-based proof of the sub-Gaussian norm concentration inequality, our method does not rely on the union bound and promises a tighter concentration bound.

Paper Structure

This paper contains 4 sections, 6 theorems, 41 equations.

Table of Contents

  1. Sub-Gaussian Norm Concentration Inequality
  2. New Proof based on Averaged Moment Generating Function
  3. Generalization to Sub-Gaussian Random Matrices
  4. Conclusion

Key Result

Theorem 1

For a sub-Gaussian vector $X\in\mathbb{R}^n$ with variance proxy $\sigma^2$, there exist constants $C_1$, $C_2$ such that, for any $\delta\in(0,1)$,

Theorems & Definitions (13)

  • Definition 1.1
  • Theorem 1
  • Definition 2.1: AMGF
  • Lemma 2.1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 3.1
  • ...and 3 more