Let $X\_1,\ldots,X\_n$ be $\{0,1\}$-valued random variables drawn from some joint distribution. Let $\tilde X\_1,\ldots,\tilde X\_n$ be their *independent version*: $\mathbb{E}X\_i=\mathbb{E}\tilde X\_i$ for each $1\le i\le n$ and the $\tilde X\_i$ are mutually independent (as well as being independent of the $X\_i$'s). I would like to compare
$M:=\mathbb{E}\max\_{1\le i\le n}X\_i$
and
$\tilde M:=\mathbb{E}\max\_{1\le i\le n}\tilde X\_i$.
It's obvious that $M$ can be made arbitrarily small, while $\tilde M$ is arbitrarily close to $1$, so in general, there can be no bound on $\tilde M/M$. But it seems that in the reverse situation, one should be able to say something. Is some bound of the form
$$ M \le c\tilde M $$
known, where $c$ is an absolute constant? If such a thing is impossible (what's the counterexample?), what's the best dependence on $n$ in the bound?



Max decoupling inequality
Yes, this inequality holds with
$$c:=\frac e{e-1}.$$


Indeed, let $A\_i:=\{X\_i=1\}$. Then
$$M=P\Big(\bigcup\_i A\_i\Big)\le\min(s,1)\le c(1-e^{-s}),$$
where $s:=\sum\_i P(A\_i)$. On the other hand,
$$\tilde M=1-\prod\_i(1-P(A\_i))
\ge1-\prod\_i\exp(-P(A\_i))=1-e^{-s}.$$
So, $M\le c\tilde M$.


