Exact converses to a reverse AM--GM inequality, with applications to sums of independent random variables and (super)martingales

TL;DR

This paper derives exact probabilistic bounds related to the ratio of the arithmetic to geometric mean of positive random variables, providing new insights and improvements to classical inequalities with applications to sums of independent variables and martingales.

Contribution

It introduces exact converse bounds to a reverse AM-GM inequality, extending classical probability inequalities and applying them to various probabilistic models.

Findings

Derived exact tail probability bounds for the ratio of arithmetic to geometric means.

Established convergence of the mean ratio to 1 as the ratio of means approaches 1.

Improved classical inequalities like Markov, Bernstein--Chernoff, and Hoeffding bounds.

Abstract

For every given real value of the ratio $\mu:=A_X/G_X>1$ of the arithmetic and geometric means of a positive random variable $X$ and every real $v>0$, exact upper bounds on the right- and left-tail probabilities $\mathsf{P}(X/G_X\ge v)$ and $\mathsf{P}(X/G_X\le v)$ are obtained, in terms of $\mu$ and $v$. In particular, these bounds imply that $X/G_X\to1$ in probability as $A_X/G_X\downarrow1$. Such a result may be viewed as a converse to a reverse Jensen inequality for the strictly concave function $f=\ln$, whereas the well-known Cantelli and Chebyshev inequalities may be viewed as converses to a reverse Jensen inequality for the strictly concave quadratic function $f(x) \equiv -x^2$. As applications of the mentioned new results, improvements of the Markov, Bernstein--Chernoff, sub-Gaussian, and Bennett--Hoeffding probability inequalities are given.