Consider some collection of weakly dependent Gaussians $\{w\_i\}$ with a uniform bound of $r$ on the magnitude of their covariances. Are there any bounds or techniques towards:
$$E[\inf\_i|w\_i|] \le f(r)$$ 
for some function $f$?


Alternatively--this would be even better--are there any comparison theorems such as:
if for two Gaussian processes $\{w\_i\}$ and $\{z\_i\}$, if $E[|w\_i - w\_j|^2] \le E[|z\_i-z\_j|^2]$, then we can say something about $E[\inf\_i|w\_i|]$ relative to $E[\inf\_i |z\_i|]$?


(Of course, if the inf is replaced with a sup, we have Sudakov-Fernique. But, replacing the sup with an inf makes this object seem quite different; am I being silly?)



Infimum of weakly dependent Gaussian process?
I haven't come across this type of problem even though I have been very interested lately in Gaussian processes and read a few books and references. Still, I am not an expert.


I propose a bound in an overly simplified setting that is nearly sharp at its endpoints. I also started studying the possibility to make a comparison lemma. I'll keep you posted if I find something in this direction. 


To simplify things a bit, first assume that $w$ is a centred Gaussian vector with $\operatorname{Var}(w\_i)=1$ for all $i$ and $\mathbb{E}(w\_iw\_j)=r>0$ for all $i$ different from $j$.Say that this vector has cardinality $n$. $Z$ denotes a one-dimensional standard Gaussian independent of all other r.v.'s appearing. I will also assume that all the elements I use (probability spaces, etc.) were well-defined. 


First, using Berman's lemma, there exists an isotropic Gaussian vector $\tilde w$ such that 
$$
\mathbb{E}(\min\_{i\le n} \vert w\_i\vert )= \mathbb{E}\left(\min\_{i\le n} \vert (1-r)^{1/2} \tilde w\_i + r^{1/2}Z\vert \right).
$$
Using triangular inequality, 
\begin{align\*}
\mathbb{E}(\min\_{i\le n} \vert w\_i\vert )&\le \mathbb{E}\left( (1-r)^{1/2} \min\_{i\le n}\vert\tilde w\_i \vert + \vert r^{1/2}Z\vert \right)\\
&\le (1-r)^{1/2} \int\_0^\infty \mathbb{P} \left(\min\_{i\le n}\vert\tilde w\_i \vert>u\right)du + \left(\frac{2r}{\pi}\right)^{1/2} \\
&\le (1-r)^{1/2} \int\_0^\infty \left[\mathbb{P} (\vert\tilde w\_i \vert>u)\right]^ndu + \left(\frac{2r}{\pi}\right)^{1/2} \\
&\le (1-r)^{1/2} \int\_0^\infty \frac{1}{u^n} \exp\left(-\frac{1}{2}nu^2\right) du + \left(\frac{2r}{\pi}\right)^{1/2}\\
&\le (1-r)^{1/2} \left(\frac{2}{\pi}\right)^{n/2} 2^{-n/2-1/2} n^{(n-1)/2} \ \Gamma(1/2 -n/2) + \left(\frac{2r}{\pi}\right)^{1/2},
\end{align\*}
where Mill's inequality was used in the fourth line.


I believe that the general case will be extremely more complicated. 
May I ask in what context this question came up ? 


Edit: I tried to develop a Sudakov-Fernique type of theorem. 


The idea is to rely on the smooth min function, construct an interpolation between the two Gaussians you want to compare and study the derivative. 


Formally, I will set the function $$F\_\beta(x):= -\beta^{-1} \log\left(\sum\_{i=1}^n \exp(-\beta \vert X\_i \vert)\right).$$ 
Then, given two centred multivariate normal random vectors X and Y with constant variance equal to 1, you can construct
$$
Z\_t=\sqrt{1-t}X +\sqrt{t} Y
$$
and set $\phi(t)= \mathbb{E}(F\_\beta(Z\_t))$.


To be able to compare the two variables, one should find the sign derivative of $\phi(t)$. We further have 


$$
\phi'(t)= \mathbb{E}\left(\sum\_{i=1}^n \frac{\partial F\_\beta(Z\_t) }{\partial x\_i} \left(\frac{Y\_i}{2\sqrt{t}}-\frac{X\_i}{2\sqrt{1-t}}\right)\right).
$$
And we can use the property that for $f$ sufficiently smooth
$$
\mathbb{E}\left(X\_i f(X)\right) = \sum\_{j=1}^n \mathbb{E}\left(X\_i X\_j\right) \mathbb{E}\left(\frac{\partial F\_\beta(X) }{\partial x\_i} \right).
$$
Then, if am not mistaken after some quite tedious calculations, it holds
$$
\phi'(t)= \frac{1}{2} \sum\_{1\le i,j\le n} \mathbb{E}\left( \frac{\exp(-\beta(\vert Z\_{t,i}\vert+\vert Z\_{t,j}\vert)}{(\sum\_l\exp(-\beta(\vert Z\_{t,l}\vert ))²} sign(Z\_{t,i})\times sign(Z\_{t,j}) (\sigma\_{i,j}^Y -\sigma\_{i,j}^X) \right),
$$
where $\sigma\_{i,j}^Y$ is the covariance between elements $i$ and $j$ in the r.v. $Y$. 


Should this last quantity always have the same sign, you would be able to prove an inequality. I am still stuck at this stage. Maybe some reader will figure a way out. The fact that we use the smooth min is not so important as the limit in $\beta$ will finally be taken. 


