Recovering a Gaussian distribution from its minimum

TL;DR

This paper investigates how to recover the covariance matrix of a Gaussian vector from the distribution of its minimum component, linking the problem to geometric identification via the circular radon transform.

Contribution

It introduces a novel approach connecting Gaussian covariance recovery to geometric problems in the circular radon transform framework.

Findings

Established conditions for covariance matrix identification from minimum distribution.

Linked the problem to geometric identification in the circular radon transform.

Provided theoretical insights into the uniqueness of the recovery process.

Abstract

Let $X=(X_1,X_2, X_3)$ be a Gaussian random vector such that $X\sim \mathcal{N} (0,\Sigma)$. We consider the problem of determining the matrix $\Sigma$, up to permutation, based on the knowledge of the distribution of $X_{\mathrm{min}}:=\min(X_1, X_2, X_3)$. Particularly, we establish a connection between this identification problem and a geometric identification problem in the context of the theory of the circular radon transform.