arXiv:2606.06384v2 Announce Type: replace Abstract: The sub-Gaussian parameter (also called the variance proxy) of a mean-zero random variable $X$ is defined as $\xi^2_\star = \sup_{\lambda \in \mathbb{R}} L(\lambda)$ where $L(\lambda) = \frac{2}{\lambda^2} \log \mathbb{E} e^{\lambda X}$ is a weighted cumulant generating function. We study the estimation of $\xi^2_\star$ and prove that the minimax risk is governed by a non-increasing function $\delta_P(C) = \sup_{|\lambda| \geq C} L(\lambda) - \sup_{|\lambda| \leq C} L(\lambda)$ which captures the influence of the tail behavior of the distribution $P$. Over the class of distributions with $\delta_P \leq r$ for a non-increasing function $r$, the minimax risk is, up to a multiplicative constant, lower bounded by $r(\sqrt{\log n}) + n^{-1/2}$ and upper bounded by $r((\log n)^{1/2-\varepsilon}) + n^{-1/2 + \varepsilon}$ for any $\varepsilon > 0$. Our estimator for the upper bound is based on constrained maximization of the empirical analogue of $L$. In addition to being almost minimax optimal and adaptive, we further prove that the estimator is asymptotic normal under suitable conditions and that, if the underlying distribution is not sub-Gaussian, the estimator diverges with a rate determined by the heaviness of the distributional tail.