Exact upper and lower bounds on the difference between the arithmetic and geometric means

Let $X$ denote a nonnegative random variable with $\mathsf{E} X<\infty$. Upper and lower bounds on $\mathsf{E} X-\exp\mathsf{E}\ln X$ are obtained, which are exact, in terms of $V_X$ and $E_X$ for the upper bound and in terms of $V_X$ and $F_X$ for the lower bound, where $V_X:=\mathsf{Var}\sqrt X$, $E_X:=\mathsf{E}\big(\sqrt X-\sqrt{m_X}\,\big)^2$, $F_X:=\mathsf{E}\big(\sqrt{M_X}-\sqrt X\,\big)^2$, $m_X:=\inf S_X$, $M_X:=\sup S_X$, and $S_X$ is the support set of the distribution of $X$. Note that, if $X$ takes each of distinct real values $x_1,\dots,x_n$ with probability $1/n$, then $\mathsf{E} X$ and $\exp\mathsf{E}\ln X$ are, respectively, the arithmetic and geometric means of $x_1,\dots,x_n$.Comment: 8 pages; to appear in the Bulletin of the Australian Mathematical Society. Version 2: the condition that the random variable X is nonnegative was missing in the abstrac

Convex cones of generalized multiply monotone functions and the dual cones

Let $n$ and $k$ be nonnegative integers such that $1\le k\le n+1$. The convex cone $\mathcal{F}_+^{k:n}$ of all functions $f$ on an arbitrary interval $I\subseteq\mathbb{R}$ whose derivatives $f^{(j)}$ of orders $j=k-1,\dots,n$ are nondecreasing is characterized in terms of extreme rays of the cone $\mathcal{F}_+^{k:n}$. A simple description of the convex cone dual to $\mathcal{F}_+^{k:n}$ is given. These results are useful in, and were motivated by, applications in probability. In fact, the results are obtained in a more general setting with certain generalized derivatives of $f$ of the $j$th order in place of $f^{(j)}$. Somewhat similar results were previously obtained in the case when the left endpoint of the interval $I$ is finite, with certain additional integrability conditions; such conditions fail to hold in the mentioned applications.Comment: Version 2: More applications given; two typos fixe

Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above

Let $(S_0,S_1,...)$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-algebras $H_{\le0},H_{\le1},...$, with $S_0\le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that $X_i\le d$ and $\mathsf {Var}(X_i|H_{\le i-1})\le \sigma_i^2$ a.s. for every $i=1,2,...$, where $d>0$ and $\sigma_i>0$ are non-random constants. Let $T_n:=Z_1+...+Z_n$, where $Z_1,...,Z_n$ are i.i.d. r.v.'s each taking on only two values, one of which is $d$, and satisfying the conditions $\mathsf {E}Z_i=0$ and $\mathsf {Var}Z_i=\sigma ^2:=\frac{1}{n}(\sigma_1^2+...+\sigma_n^2)$. Then, based on a comparison inequality between generalized moments of $S_n$ and $T_n$ for a rich class of generalized moment functions, the tail comparison inequality \mathsf P(S_n\ge y) \le c \mathsf P^{\mathsf Lin,\mathsf L C}(T_n\ge y+\tfrach2)\quad\forall y\in \mathbb R is obtained, where $c:=e^2/2=3.694...$, $h:=d+\sigma ^2/d$, and the function $y\mapsto \mathsf {P}^{\mathsf {Lin},\mathsf {LC}}(T_n\ge y)$ is the least log-concave majorant of the linear interpolation of the tail function $y\mapsto \mathsf {P}(T_n\ge y)$ over the lattice of all points of the form $nd+kh$ ($k\in \mathbb {Z}$). An explicit formula for $\mathsf {P}^{\mathsf {Lin},\mathsf {LC}}(T_n\ge y+\tfrac{h}{2})$ is given. Another, similar bound is given under somewhat different conditions. It is shown that these bounds improve significantly upon known bounds.Comment: Published at http://dx.doi.org/10.1214/074921706000000743 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org