# Convexity of the variance of a function depending on random variables

here is my question:

I have a function $f(x,\epsilon_1, \dots, \epsilon_n)$ that depends on a decision $x$ I make and a certain amount of random variables $\epsilon_i$.

I define the two following functions:

$g(x) = \mathbf{Var}[(f(x,\epsilon_1, \dots, \epsilon_n))^2]$

and

$h(x) = \mathbf{Var}[|f(x,\epsilon_1, \dots, \epsilon_n))|]$.

I would like to know if the following implication holds:

$$g \text{ is convex} \Rightarrow h \text{ is convex }.$$

• no; counterexample: $f(x,\epsilon_1,\ldots\epsilon_n)=|x|^{3/8}u(\epsilon_1,\ldots\epsilon_n)$; $g(x)=|x|^{3/2}\,\rm{var}\,u$ is a convex function of $x$, while $h(x)=|x|^{3/4}\,{\rm var}\,u$ is not – Carlo Beenakker May 15 at 19:28
• typo: the first ${\rm var}\,u$ should read ${\rm var}\,u^2$, the second ${\rm var}\,|u|$ --- makes no difference for the convexity, of course. – Carlo Beenakker May 15 at 19:56