Odd central moments of unimodal distributions (Q1181085)

The main result of the paper is the following. Let \(F\) be a unimodal probability measure with finite mean \(\mu_ F\). Suppose that \(F\) is non- symmetric and that \[ \inf\{y\in(0,y_{max}):\;h_ F(y)<\mu_ F\}\geq \sup\{y\in(0,y_{max}):\;h_ F(y)>\mu_ F\}, \] where \[ h_ F(y)=(\inf\{x\in R:\;f(x)>y\}+\sup\{x\in R:\;f(x)>y\})/2 \] (\(f\) is the right continuous version of the density of \(F\)). Then, for \(k=1,2,3,\dots\) \[ \int_ R (x-\mu_ F)^{2k+1} F(dx)>0 \] whenever the left hand side is well-defined. The main idea is a new decomposition result for unimodal distributions.