In this paper, we derive some upper and lower bounds and inequalities for the total variation distance (TVD) and the Kullback-Leibler divergence (KLD), also known as the relative entropy, between two probability measures $\mu$ and $\nu$ defined by $$ D_{\mathrm{TV}} ( \mu, \nu ) = \sup_{B \in \mathcal{B} (\mathbb{R}^n)} \left| \mu(B) - \nu(B) \right| \quad \text{and} \quad D_{\mathrm{KL}} ( \mu \, \| \, \nu ) = \int_{\mathbb{R}^n} \ln \left( \frac{d\mu(x)}{d\nu(x)} \right) \, \mu(dx) $$ correspondingly when the dimension $n$ is high. We begin with some elementary bounds for centered elliptical distributions admitting densities and showcase how these bounds may be used by estimating the TVD and KLD between multivariate Student and multivariate normal distribution in the high-dimensional setting. Next, we show how the same approach simplifies when we apply it to multivariate Gamma distributions with independent components (in the latter case, we only study the TVD, because KLD may be calculated explicitly, see [1]). Our approach is motivated by the recent contribution by Barabesi and Pratelli [2].

翻译：本文针对高维情形（维度$n$较大时），推导了定义如下的两个概率测度$\mu$与$\nu$之间的总变差距离（TVD）与Kullback-Leibler散度（KLD，亦称相对熵）的若干上下界与不等式： $$ D_{\mathrm{TV}} ( \mu, \nu ) = \sup_{B \in \mathcal{B} (\mathbb{R}^n)} \left| \mu(B) - \nu(B) \right| \quad \text{与} \quad D_{\mathrm{KL}} ( \mu \, \| \, \nu ) = \int_{\mathbb{R}^n} \ln \left( \frac{d\mu(x)}{d\nu(x)} \right) \, \mu(dx) $$ 我们首先针对具有密度函数的中心椭圆分布给出一些基本界，并通过估计高维情形下多元Student分布与多元正态分布之间的TVD与KLD，展示这些界如何应用。接着，我们将相同方法应用于具有独立分量的多元Gamma分布（在后一情形中，我们仅研究TVD，因为KLD可显式计算，参见文献[1]），并展示该方法的简化过程。我们的研究动机源于Barabesi与Pratelli最近的贡献[2]。