Asymptotics of the minimum of Binomial random variables Let $Y=\min\{X_1, X_2 \cdots X_k\}$ be the minimum of $k$ iid Binomial $(n,1/2)$ random variables.
I'm interested in the asymptotics of $Y$ (distribution, or mean and variance) for large $n$ and $k = 2^{ \beta n }$, with $\beta < 1/2$.
Any suggestions or pointers would be appreciated.
 A: This attempts to get the asymptotical behaviour of $P(Y)$, making some adaptations (and, I think, some simplifications and improvements) to this answer in other SE site, which corresponds to essentially this problem with $\beta < 1/2$. 
We seek to find some $d$ such that $P(X_i\le d) \approx 1/k$, so that  $P(Y>d) \approx (1-1/k)^k \to e^{-1}$.
We expect (hope) that $d/n$ will be asymptotically constant, as $n\to\infty$.
Let's use the approximation $\log_2 \binom{n}{d} \approx n \, h(d/n)$, where $h()$ is the binary entropy function, so
$$P(X_i=d)\approx 2^{-n(1-h(d/n))} \tag1$$
Noticing that for $\delta \ll x$ we have $h(x+\delta) \approx h(x) - \delta \log_2(\frac{x}{1-x})$, or 
$$h\left(\frac{d-j}{n}\right)\approx h(d/n) + \log_2\left(\frac{d}{n-d}\right) \frac{j}{n} \tag2$$
then
$$P(X_i=d-j)\approx  P(X_i=d) \left(\frac{d}{n-d}\right)^j \tag3$$
and
$$P(X_i \le d) = \sum_{j=0}^d P(X_i=d-j) \approx  P(X_i=d) \left(\frac{n-d}{n-2d}\right) \tag 4$$
Plugging $(1)$ into $(4)$ and equating it to $1/k= 2^{-\beta n}$, calling $t=d/n$, we get
$$ \log_2\left(\frac{1- 2 t}{1-t}\right)=n \left(h(t) - 1 +\beta \right) \tag{5}$$
If we impose that $t$ is (asympotically) constant wrt $n$, then this implies that (asympotically) this constant (which we call $\hat{t}$) must be a root of
$$ h(t)=1-\beta \tag{6}$$
In this case, it should be the lower ($t<1/2$) root. For example, for $\beta=1/4$ we
get $\hat{t}=0.2145017$. 
So we take $d = n \hat{t} $ , rounded to the nearest integer.
This implies (details here) that the distribution of $Y$ corresponds to a Gumbel distribution, with mean tending asymptotically to $d$ (hence growing linearly with $n$) and constant variance. Hence $Y$ is asymptotically concentrated around the root of $(6)$.
