Let $Z\_1,\dots,Z\_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z\_1,\dots,Z\_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in general?



Log concavity of the maximum of dependent Gaussians
$\newcommand{\R}{\mathbb{R}}
\renewcommand{\P}{\operatorname{\mathsf P}}
\newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}}$ 


This is false in general. E.g., let $Z\_1=U$ and $Z\_2=|V|\,\text{sign}\,U$, where $U,V$ are iid standard normal random variables. Let $F$ be the cdf of $\max(Z\_1,Z\_2)$ and $L:=\ln F$. 
Let also $\Phi$ denote the standard normal cdf. 
Then $Z\_1$ and $Z\_2$ are each Gaussian and for $x\ge0$
\begin{align\*}
 F(x)&=\P(U<x,|V|\,\text{sign}\,U<x) \\ 
 &=\P(U<0)\P(|V|>-x)+\P(0<U<x)\P(|V|<x) \\ 
 &=\tfrac12+[\Phi(x)-\Phi(0)][\Phi(x)-\Phi(-x)], 
\end{align\*}
which yields $L''(0+)=4/\pi>0$, so that $F$ is not log concave. 


However, if $Z\_1,\dots,Z\_n$ are *jointly* normal, then for $x\in\R$
\begin{equation}
 F(x)=\int\_{\R^n}f(z\_1,\dots,z\_n)\ii{z\_1<x}\cdots\ii{z\_n<x}\,dz\_1\cdots dz\_n,
\end{equation}
where $F$ is the cdf of $\max(Z\_1,\dots,Z\_n)$, $f$ is the pdf of $(Z\_1,\dots,Z\_n)$, and $\ii\cdot$ denotes the indicator. 
So, by the Prékopa–Leindler theorem (see Section [Applications in probability and statistics](https://en.wikipedia.org/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality)), $F$ is log concave -- since $f$ is log concave, $\ii{z\_i<x}$ is log concave in $(z\_i,x)$ for each $i$, and the product of log-concave functions is log concave. 


