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A general decoupling inequality for finite Gaussian vectors

Michel Weber

TL;DR

This work addresses when a decoupling inequality holds for finite Gaussian vectors by deriving a general bound for $|\,\mathbb{E}(\prod_{i=1}^n f_i(X_i))|$ in terms of $L^p$ norms. It develops a method based on Brascamp–Lieb inequalities and simultaneous diagonalization to produce an explicit, potentially optimal $p$-region $S$ for which the bound is valid, with a constant that depends on the variances $\sigma_i^2$, the determinant $\det(C)$, and a spectrum $(\xi_i)$ arising from the diagonalization. A key contribution is the reformulation of the region in terms of eigen-structure via $\lambda_j=p\xi_j$, linking $\det(pI(\underline{\gamma})-C)$ to a computable spectral characterization. The results provide a concrete, computable criterion for decoupling in Gaussian settings, with potential applications in probability, statistics, and related numerical methods.

Abstract

We use Matrix Analysis to prove a general decoupling inequality for finite Gaussian vectors, in identifying a new region of the inherent $p$ exponent, for the validity of this one.

A general decoupling inequality for finite Gaussian vectors

TL;DR

This work addresses when a decoupling inequality holds for finite Gaussian vectors by deriving a general bound for in terms of norms. It develops a method based on Brascamp–Lieb inequalities and simultaneous diagonalization to produce an explicit, potentially optimal -region for which the bound is valid, with a constant that depends on the variances , the determinant , and a spectrum arising from the diagonalization. A key contribution is the reformulation of the region in terms of eigen-structure via , linking to a computable spectral characterization. The results provide a concrete, computable criterion for decoupling in Gaussian settings, with potential applications in probability, statistics, and related numerical methods.

Abstract

We use Matrix Analysis to prove a general decoupling inequality for finite Gaussian vectors, in identifying a new region of the inherent exponent, for the validity of this one.

Paper Structure

This paper contains 7 sections, 6 theorems, 52 equations.

Table of Contents

  1. Result
  2. Preparation
  3. Brascamp-Lieb's inequality
  4. Estimates of $\boldsymbol{E_A}$
  5. Proof
  6. Discussion
  7. Concluding Remarks

Key Result

Theorem 1.1

Let $X=\{X_i, 1\le i\le n\}$ be a centered Gaussian vector with invertible covariance matrix $C$, and let ${\sigma}_i^2={\mathbb E} X_i^2>0$ for each $1\le i\le n$, $\underline{{\gamma}}=({\sigma}^2_1, \ldots ,{\sigma}^2_n)$. Let ${\beta}\ge 1$ be chosen so that $\bar{{\beta}}:= \frac{(\max {\sigma} where $p(X)$ is the decoupling coefficient of $X$ defined by Then for any complex-valued measurabl

Theorems & Definitions (6)

  • Theorem 1.1: W3, Th. 2.3
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2: W3, Prop. 3.4
  • Lemma 3.1
  • Lemma 4.1