[This post](https://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 $P(\max\_{1\leq i\leq m} X\_i > c\_{m,n})\to 0$ 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 $n$. My gut also tells me that $c\_{m,n}$ will depend on some function of the $p\_i$'s (in addition to $m$ and $n$).



Tail bound for maximum of independent (but not identical) binomial random variables
You can derive some conditions applying the Bernstein's trick. For any positive $t$, we have


$$
P\left( \max\_{i\le m}X\_i > c\_{n,m}\right) = P\left( \exp\{t\max\_{i\le m}X\_i\} > e^{t c\_{n,m}}\right) \le e^{-t c\_{n,m}}E\left[ \max\_{i\le m} e^{tX\_i}\right]
$$
Since $X\_i$ follows a binomial distribution
$$
E\left[ e^{tX\_i}\right] = (1-p\_i +p\_ie^{t})^n \le \exp\{ p\_i(e^t-1)n\}.
$$
Bounding the maximum by the sum and denoting by $p\_m$ the largest $p\_i$, we have
$$
P\left( \max\_{i\le m}X\_i > c\_{n,m}\right) \le m\exp\{ p\_m(e^t-1)n -tc\_{n,m}\}.
$$
Now, you can minimize on $t$ the right-hand side of the above inequality and finally impose some conditions over $\max\_i p\_i$, $m$ and $n$.
This method is very useful when you know the distribution's MGF and have independence.


A wonderful and useful reference for concentration Inequalities and tail bounds like these is the book:


Concentration Inequalities: A Nonasymptotic Theory of Independence. S. Boucheron, G. Lugosi and P. Massart.


<https://books.google.com.br/books/about/Concentration_Inequalities.html?id=ZG6OjgEACAAJ&redir_esc=y&hl=en>


Hope it helps.


