Abstract:

We derive upper and lower bounds for the total variation distance (TVD) and the Kullback–Leibler divergence (KLD), also known as the relative entropy, between two probability measures μ and ν defined by

\[ D_{TV}(\mu, \nu) = \sup_{B \in \mathcal{B}(\mathbb{R}^n)} |\mu(B) - \nu(B)| \]

and Kullback-Leibler Divergence

\[ D_{KL}(\mu \parallel \nu) = \int_{\mathbb{R}^n} \ln \left( \frac{d\mu(x)}{d\nu(x)} \right) \mu(dx) \]

in high-dimensional settings. We establish elementary bounds for centered elliptical distributions and apply them to multivariate Student t-distributions versus multivariate normal distributions. For the latter case, we provide an exact formula via integration over the unit sphere, computable bounds in terms of special functions, and concentration-based bounds in which the worst-case parameters are replaced by effective averaged quantities, up to exponentially small exceptional-set corrections. We also apply our approach to multivariate Gamma distributions with independent components. Our work is motivated by the recent contribution of Barabesi and Pratelli (2024)

2010 AMS Mathematics Subject Classification: Primary 60E15; Secondary 62H05.

Keywords and phrases: total variation distance, Kullback--Leibler divergence, multivariate Student distribution, high-dimensional statistics, Gamma distribution.