Let $X_i \ge 0, I=1,\ldots,N$ be a set of random variables that are identically distributed but dependent. We have a bound $P(X_i\ge L)\le f(L)$, for some known $f$. Is it possible to set an upper bound on $P(\max_i X_i > L)$? My conjecture is that the best possible bound for arbitrary sets of random variables is the one that we'd compute for iid rvs:
\begin{align}
P(\max_i X_i > L)=1- \prod_i [1-P(X_i>L)]\le 1 -[1-f(L)]^N
\end{align}
Best Answer
I would have thought an upper bound might be $\mathbb P(\max\limits_i X_i\gt L) \le N \mathbb P(X_i\gt L)$. It is an upper bound since you need at least one of the $N$ events of $X_i >L$ to have $\max\limits_i X_i\gt L$, and is tight if at most one ever meets that condition.
This is the first term in the expansion of your $1 -[1-f(L)]^N$. Let's take an example where my upper bound is tight and yours is not an upper bound.