Upper and lower bounds on TVD and KLD between centered elliptical distributions in high-dimensional setting
Pavel Ievlev, Timofei Shashkov · Probability and Mathematical Statistics · 2026
In this paper, we derive some upper and lower bounds and inequalities for thetotal variation distance (TVD) and the Kullback-Leibler divergence (KLD), alsoknown as the relative entropy, between two probability measures $\mu$ and$ u$ defined by$$D_{\mathrm{TV}} (\mu, u) = \sup_{B \in \mathcal{B} (\mathbb{R}^n)}\left| \mu(B) - u(B) \right|\quad \text{and} \quadD_{\mathrm{KL} ( \mu \, \| \, u ) = \int_{\mathbb{R}^n}\ln \left( \frac{d\mu(x)}{d u(x)} \right) \, \mu(dx)$$correspondingly when the dimension $n$ is high. We begin with someelementary bounds for centered elliptical distributions admitting densities andshowcase how these bounds may be used by estimating the TVD and KLD betweenmultivariate Student and multivariate normal distribution in thehigh-dimensional setting. Next, we show how the same approach simplifies when weapply it to multivariate Gamma distributions with independent components (in thelatter case, we only study the TVD, because KLD may be calculated explicitly,see [1]). Our approach is motivated by the recent contribution byBarabesi and Pratelli [2].