A Gamma envelope and sharp moment inequalities for Gaussian quadratic forms
Abstract
We study extremal absolute central moments of Gaussian quadratic forms under a fixed Frobenius norm. For a nonzero real symmetric matrix M and G∼ N(0,In), we construct an explicit centered difference of Gamma variables with the same mean, variance, and third centered moment as GTMG−trM. After normalization, replacing the quadratic form by this Gamma difference does not decrease 𝔼 f for every C2 test function f such that f'' is convex and f, f', and f'' have polynomial growth. In particular, it gives an explicit upper bound for every absolute moment of order p≥3. We then prove that, for every p≥4, this bound is maximized by the centered square g2−1 of a single standard Gaussian variable g. The resulting sharp inequality is |GTMG−trM|p ≤ |g2−1|p |M|F, with equality if and only if M has rank one.
Related
- Moment-Orthogonal Quadratic-Envelope E-Processes for Anytime-Valid Location Inference
- Computing the convex envelope of bivariate piecewise linear-quadratic functions in linear time
- Moment-optimal Gamma approximation of Bessel Process exit times
- Matrix-Free Moment-Matching Method for Reduced-Order Modeling of Quadratic-Bilinear Descriptor Systems with Multiple Inputs
- The first moment of quadratic Dirichlet L-functions in the even hyperelliptic ensemble
BibTeX
@misc{pang2026gamma,
title = {{A Gamma envelope and sharp moment inequalities for Gaussian quadratic forms}},
author = {Zhekai Pang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05914},
eprint = {2609.05914},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05914},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Gamma envelope and sharp moment inequalities for Gaussian quadratic forms
AU - Zhekai Pang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05914
UR - https://arxiv.org/abs/2609.05914
AB - We study extremal absolute central moments of Gaussian quadratic forms under a fixed Frobenius norm. For a nonzero real symmetric matrix M and G∼ N(0,I_n), we construct an explicit centered difference of Gamma variables with the same mean, variance, and third centered moment as G^TMG−trM. After normalization, replacing the quadratic form by this Gamma difference does not decrease 𝔼 f for every C^2 test function f such that f'' is convex and f, f', and f'' have polynomial growth. In particular, it gives an explicit upper bound for every absolute moment of order p≥3. We then prove that, for every p≥4, this bound is maximized by the centered square g^2−1 of a single standard Gaussian variable g. The resulting sharp inequality is |G^TMG−trM|_p ≤ |g^2−1|_p |M|_F, with equality if and only if M has rank one.
ER -