Bound of supremum of identically distributed but dependent random variables

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}

Solution 1:

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.

• Suppose $N=2$ and and $L=\frac12$ $\mathbb P(X_1=1 \text{ and } X_2=0) = \mathbb P(X_1=0 \text{ and } X_2=1) =\frac12$
• Then $X_1$ and $X_2$ are identically distributed and $\mathbb P(X_i\gt \frac12)=\frac12$
• and $\mathbb P(\max\limits_i X_i\gt \frac12)=1 = 2\times \mathbb P(X_i\gt \frac12) > 1- (1-\mathbb P(X_i\gt \frac12))^2 = \frac34$