Gaussian maxima reach highest likelihood in regular simplex pattern
Stochastic Domination of Gaussian Maxima by the Regular Simplex
Information Theory
Summary
This paper studies how the maximum values among correlated Gaussian variables compare to a special, balanced arrangement called a regular simplex. The authors show that the chance that all variables stay below a certain level is always at least as big as for the regular simplex configuration, and the only time they are equal is when the variables have a very specific pattern of negative correlations. This result confirms a conjecture about Gaussian probabilities linked to geometric shapes and helps understand optimal signal designs in noisy communication systems.
What this means in practice
- •For communications engineers: Optimize signal constellations with equal-energy signals in Gaussian noise for reliable message identification at given false-alarm rates.
- •For statistical signal processors: Design optimal hypothesis tests involving multiple equally likely Gaussian signals to maximize correct detection probabilities.
A theory result. No direct application yet.
Authors
Abhijeet Mulgund
Abstract
Let $n\ge2$, and let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector with $\mathrm{Var}(X_i)=1$ for every $i$. Let $Z_1,\ldots,Z_n$ be independent standard Gaussians, and put $\overline{Z}=(Z_1+\cdots+Z_n)/n$. We prove $\mathbb{P}\{\max_i X_i\le t\}\ge\mathbb{P}\{\sqrt{n/(n-1)}\,\max_i(Z_i-\overline{Z})\le t\}$ for every $t\in\mathbb{R}$, and for each fixed $t>0$ equality holds only when $\mathrm{Cov}(X_i,X_j)=-1/(n-1)$ for all $i\ne j$. The right side is the distribution function of the maximum of the regular simplex vector. Equivalently, among all simplices containing a given centered ball, the regular simplex circumscribed about the ball has the least standard Gaussian measure, as conjectured by Balitskiy, Karasev, and Tsigler. In our preceding paper we proved this comparison after both maxima are smoothed by independent Gaussian noise of variance $1/(n-1)$, which suffices for the Weak Simplex Conjecture; here we remove the smoothing, which is what probabilities at a single threshold require. As an application we consider $n$ equally likely signals of equal energy in Gaussian noise, where the transmitter may also send nothing. At every positive false-alarm level, and for every law of a common nonnegative random amplitude not concentrated at zero, the regular simplex uniquely maximizes the average probability of correct identification whenever the signal dimension is at least $n-1$. A Lean formalization is available at https://github.com/abhmul/full-simplex-conjecture-lean.