I'm trying to figure out a lower bound on the variance of the follow: $\max_i(x_1,x_2,...x_m)$. Where $x_i, i \in m$ are independent random variables that shares the same variance $\sigma_x$.
I know we can upper bound the above quantity to be: $\text{Var}[max_i(x_1,x_2,...x_m)] \leq \sum^m_i \text{Var}[x_i] $, but what about a lower bound? Also, is there a guarantee that this is the tightest lower bound?
Also, can we find a bound for $\text{Var}[max_i(a_1,a_2,...a_m),max_i(b_1,b_2,...b_m),...,max_i(z_1,z_2,...z_m)] $
Unfortunately, for $m\ge 2$, there is no lower bound in terms of the variances of $X_i$ (except the trivial one).
@Michael explained that the variance may vanish as $m\to\infty$, but this can happen even for fixed $m\ge 2$. Indeed, let $\mathrm{P}(X_i=1/\sqrt{p(1-p)}) = 1- \mathrm{P}(X_i=0) = p$ so that $\mathrm{Var}(X_i) = 1$. Then, $$ \mathrm{P}\left(\max_{1\le i\le m} X_i=\frac{1}{\sqrt{p(1-p)}}\right) = 1- \mathrm{P}\left(\max_{1\le i\le m} X_i=0\right)= 1 - p^m, $$ so $$ \mathrm{Var}\left(\max_{1\le i\le m} X_i\right) = \frac{p^m(1-p^m)}{p(1-p)}\to 0, p\to 0+. $$
Under the assumption that the variables are bounded by some $C$, it may be possible to get a lower bound depending on $C$ (and $m$).