Let $S$ be the class of all $2$ by $2$ matrices of the form
$$\begin{bmatrix}
 1 & a \\
 a & 1 
\end{bmatrix},\, |a|\leq 1.$$
Is there a single matrix $M\in S$ such that for any $N\in S$ and all $x>0$ we have
$$\mathbb{P}(||X||\_2 \geq x)\geq \mathbb{P}(||Y||\_2 \geq x),$$
where $X$ and $Y$ have the Gaussian distribution with covariance matrices $M$ and $N$, respectively?
In other words, is it possible to determine which covariance structure gives the fattest tail behavior?



Stochastic domination of Gaussian random vectors
$\renewcommand{\P}{\operatorname{\mathsf P}}\newcommand{\E}{\operatorname{\mathsf E}}$The answer is no. More specifically, let $Y\_a$ be a centered Gaussian random vector with covariance matrix $\begin{bmatrix}
 1 & a \\
 a & 1 
\end{bmatrix}$. 
Then there is some $u\_1\in(0,\infty)$ such that 
\begin{equation}
 \max\_{|a|\le1}\P(\|Y\_a\|\_2 \ge u)
 =\begin{cases}
 \P(\|Y\_0\|\_2 \ge u) &\text{ if } u\le u\_1,\\
 \P(\|Y\_1\|\_2 \ge u) &\text{ if } u\ge u\_1. 
 \end{cases}
\end{equation}


This is a special case of [Theorem 1](https://link.springer.com/article/10.1007/s00440-003-0262-6), which states the following: 



> 
> For each natural $d$, the equation $\P(\chi\_d^2/d>x\_d)=\P(\chi\_{d+1}^2/(d+1)>x\_d)$ in $x\in(0,\infty)$ has a unique root $x\_d$, and we have $x\_0:=\infty>x\_1>x\_2>\cdots$; here, $\chi\_d^2$ is a random variable (r.v.) with the chi-squared distribution with $d$ degrees of freedom.  
> 
>  For any natural $n$, let $\mathcal Q\_n$ be the set of all positive semidefinite quadratic forms $Q\_n$ in centered Gaussian r.v.'s such that $\E Q\_n=1$. Then 
>  \begin{equation}
>  \min\_{Q\_n\in\mathcal Q\_n}\P(Q\_n\le x)=
>  \begin{cases}
>  \P(\chi\_n^2/n\le x)&\text{ if }x\in[0,x\_{n-1}), \\
>  \P(\chi\_d^2/d\le x)&\text{ if }d\in\{n-1,\dots,1\}\ \& \ x\in[x\_d,x\_{d-1}). 
>  \end{cases}
> \end{equation}
> 
> 
> 


