I have a sequence of independent random variables $U_1, U_2, \dots,U_N$.
Suppose $\mathbb{E}[U_i] \leq 1$ for all $i=1,\dots,N$, and let: $$M_N = \max_{i=1,\dots,N} U_i$$
It is easy to see that $\mathbb{E}[U_i] \leq N$.
My questions are:
- Is there a general tighter upper bound on the expectation of $M_N$?
- If the answer is negative, what do I need from the $U_i$ to have a tighter bound?
For instance, if the $U_i$ are Gaussian with mean $\mu_i$ and variance $\sigma$, I can use the ideas here to get the bound: $$\mathbb{E}[M_N] \leq \bar{\mu} + \sigma\sqrt{2\log(N)},$$ where $\bar{\mu} = \frac{1}{N}\sum_{i=1}^N \mu_i$.
