Stochastic Domination of Gaussian Maxima: A Resolution to the Weak Simplex Conjecture
2026-07-15 • Information Theory
Information Theory
AI summaryⓘ
The authors study the maximum value of correlated Gaussian random variables compared to independent ones. They show that if the correlations satisfy a certain condition, the maximum from the correlated variables is statistically smaller than that from independent ones. This result solves the Weak Simplex Conjecture in signal processing, proving that a regular simplex arrangement of signals is optimal for decoding in noisy communication channels. Their proof also confirms a related geometric inequality and gives a formula for the maximum number of messages that can be sent reliably under certain energy constraints.
Gaussian maximacorrelation matrixstochastic orderingWeak Simplex Conjecturemaximum-likelihood decodingadditive white Gaussian noise (AWGN)regular simplexlog-concave functionsGaussian product inequalityenergy constraint
Authors
Abhijeet Mulgund
Abstract
We prove a stochastic comparison for Gaussian maxima. Let $R$ be an $m\times m$ correlation matrix satisfying $R-\mathbf{1} \mathbf{1}^{\mathsf T}/m\succeq0$, let $X\sim\mathcal{N}(0,R)$, and let $Z_1,\ldots,Z_m$ be independent standard Gaussian random variables. Then $\max_{1\leq i\leq m}X_i \leq_{\mathrm{st}} \max_{1\leq i\leq m}Z_i$, or equivalently, $\mathbb{P}\{X_i\leq c\text{ for every }i\}\geqΦ(c)^m$ for every $c\in\mathbb{R}$. This comparison resolves the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in $\mathbb{R}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex maximizes the probability of correct maximum-likelihood decoding at every signal-to-noise ratio. It also proves the inequality asserted by the Simplex Mean Width Conjecture and gives an exact formula for the largest number of equiprobable messages that can be sent at prescribed energy and error probability by a deterministic no-feedback AWGN code under a per-codeword energy constraint. The proof combines a Gaussian product inequality for log-concave functions with an adaptive tilting argument that makes the inequality applicable to the one-sided threshold events defining the maximum.