Log concavity of the maximum of dependent Gaussians 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?
 A: $\newcommand{\R}{\mathbb{R}}
\renewcommand{\P}{\operatorname{\mathsf P}}
\newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}}$ 
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),  $F$ 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. 
