[This post](http://mathoverflow.net/questions/151033/asymptotics-of-the-maximum-of-binomial-random-variables) derived the tail bound for the maximum of independent and identically distributed binomial r.v.'s based on normal approximation. Is there a similar result in the literature for finding the bounding constant (in the post, it is $x_n$) for the case where the binomial r.v.'s may not be identically distributed? 

Mathematically, let $X_i\stackrel{indep}{\sim} Bin(n,p_i)$, $1\leq i\leq m$. 

> Is it possible to find a $c_{m,n}$, such that $P(\max_{1\leq i\leq m} X_i > c_{m,n})\to 1$ as $m,n\to\infty$, also assuming that $m=\mathcal{O}(n^r)$ for some $r>1$?

The condition at the end lets $m$ get larger at a rate faster than $n$. It also seems possible that $c_{m,n}$ will depend on sum function of the $p_i$'s