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

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.