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.
翻译:$\mu:=A_X/G_X>1美元,对正随机变量的计算和几何手段的每个实际价值,美元=A_X/G_X>1美元,每实际美元=0美元,右和左尾概率的精确上限,美元=mathsf{P}(X/G_X\ge v)美元和美元=mathsf{P}(X/G_X\le v)美元(X/G_X=1美元),对美元和美元=mathsf{G_G_X>1美元(X_X_X>1美元)(X/G_X_X>1美元),对美元和美元(x/G_X_X_Downrowl1美元),对美元和美元(xxxx_X_X_xxx_1美元),对美元,对美元和美元(xxxxxxxxxx_X_X_x1美元),对美元,对美元,对美元,对美元,对右和左尾功能功能的纠正Jen的不平等的偏差值。 Mark-zen-c,对Ben-zin-zin-zin-zin-c,对Bin-ch-c,对Bin-zin-zin-zin-zin-zin-ch),对Ben-chen的修改。