Optimal sub-Gaussian variance proxy for 3-mass distributions

TL;DR

This work develops a general, variation-based characterization of the optimal sub-Gaussian variance proxy $\sigma^2_{ ext{opt}}$ for real-valued random variables, focusing on discrete, equidistant distributions. Central to the method is the function $g_Y(\lambda;\sigma^2)=\tfrac{1}{2}\lambda^2\sigma^2-M_Y(\lambda)$ and the equation system $g_Y(\lambda;\sigma^2)=0$, $g_Y'(\lambda;\sigma^2)=0$, which reduces the search to zeros of $\lambda M_Y'(\lambda)-2M_Y(\lambda)=0$ and verification of local minimality. The paper derives a complete picture for 3-mass distributions: a phase transition at $p=1/6$ in the symmetric case where $\sigma^2_{ ext{opt}}=\operatorname{Var}[Y]$ for $p\geq 1/6$, and explicit closed forms in several asymmetric regimes, with a nonlinear-equation characterization when the central mass is large. It also proves that the discrete uniform distribution on $N$ points is strictly sub-Gaussian with $\sigma^2_{ ext{opt}}=\operatorname{Var}[X]$, and provides an open-source Python package that blends analytic and numerical techniques to compute optimal variance proxies for a wide class of distributions, aiding probabilistic tail control and Bayesian analysis. Together, these results advance discrete tail analysis and offer practical tools for sharp concentration bounds. $\sigma^2_{ ext{opt}}$ is thus connected to the cumulant structure via $M_Y$, enabling precise tailoring of concentration guarantees in discrete settings.

Abstract

We investigate the problem of characterizing the optimal variance proxy for sub-Gaussian random variables,whose moment-generating function exhibits bounded growth at infinity. We apply a general characterization method to discrete random variables with equally spaced atoms. We thoroughly study 3-mass distributions, thereby generalizing the well-studied Bernoulli case. We also prove that the discrete uniform distribution over $N$ points is strictly sub-Gaussian. Finally, we provide an open-source Python package that combines analytical and numerical approaches to compute optimal sub-Gaussian variance proxies across a wide range of distributions.

Figures (4)

• Figure 1: Probability mass function of the discrete distributions covered in the paper.
• Figure 2: Illustration of \ref{['GeneralCharac']} in the case of an asymmetric $3$-mass distribution $Y$ with parameters $p_1=0.05$ and $p_2=0.01$ (see \ref{['subsec;asymmthreepoints']}). The black curve represents function $\lambda\mapsto \lambda M_Y'(\lambda)-2M_Y(\lambda)$ of Equation \ref{['eq:sys-eq']}. The green/red box plots represent the local behavior of $\lambda\mapsto g_Y\left(\lambda;\sigma^2=\frac{M_Y'(\lambda^*)}{\lambda^*}\right)$ at each zero $\lambda^*$ of the black curve to decide if $\lambda^*$ is a local minimum of $g_Y$ (in green) or not (in red). In this example, $\mathcal{L}_c^*=\{\lambda_{c_1}\approx -5.41,\lambda_{c_2}\approx 9.09\}$ yielding $\mathcal{S}_c^*=\{s_{c_1}\approx 0.17$ and $s_{c_2}\approx 0.11\}$ while $\operatorname{Var}[Y]\approx 0.059$. Optimal variance proxy is thus $\sigma_{\mathrm{opt}}^2=s_{c_1}\approx 0.17$.
• Figure 3: Left: Symmetric 3-mass case (Section \ref{['subsec;symmthreepoints']}). Black: Optimal variance proxy $\sigma_{\mathrm{opt}}^{2}$ for $0<p<\frac{1}{6}$. Blue: variance $\operatorname{Var}[Y] = 2p$. Red: upper bound $\sigma_{1}^2(p)$ of \ref{['eq:sigma1']}. Green: upper bound $\sigma_{2}^2(p)$ of \ref{['eq:sigma2']}. Right: Asymmetric 3-mass case (Section \ref{['subsec;asymmthreepoints']}). Two different regimes depending on the relative weight of the intermediate mass.
• Figure 4: Functions $\lambda \mapsto g_{\sigma,p_1,p_2}(\lambda)$ of Equation \ref{['eq:gp12']} for $(p_1, p_2) = (0.13, 0.25)$, with $\sigma^2$ varying from the upper bound $\frac{2\sqrt{p_1p_2}}{p_3+2\sqrt{p_1p_2}}$ in dark red down to the variance $\text{Var}[Y]$ in blue (the function $g$ then becomes locally negative around $\lambda = 0_-$), while the orange curve stands for the optimal proxy variance $\sigma_{\mathrm{opt}}^2= \frac{2(p_2-p_1)}{\ln p_2 - \ln p_1}$. The intermediate curves illustrate the progressive transition from convex behavior around 0 to oscillating behavior.

Theorems & Definitions (28)

• Definition 1.1: Sub-Gaussian variables
• Proposition 2.1: Asymptotics at infinity and sufficient condition for sub-Gaussianity.
• Theorem 2.2: Characterization of the optimal variance proxy.
• Remark 2.3
• Remark 2.4
• Proposition 2.5: Case when $g_Y"$ has at most one zero on a half-line.
• Proposition 2.6: Case when $g_Y"$ has exactly two zeros on a half-line.
• Theorem 3.1: Optimal variance proxy for symmetric 3-mass distribution.
• Theorem 3.2: Optimal variance proxy for $p_3\leq 4\sqrt{p_1p_2}$, closed-form expression.
• Corollary 3.3