Local behavior of the Vandermonde convolution

Question

An interesting combinatorial identity is the Vandermonde convolution identity:

$$ \sum_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$

which can be proved by considering the coefficients in $(x+1)^{n+m} = (x+1)^n (x+1)^m$. I am interested in the behavior of the summands: which summands among them contribute the most?.

More specifically, I'd like to look at the behavior of the function

$$ F(k) = \frac{{n\choose k}{m\choose s-k}}{{n+m \choose s}}. $$

In a specific probabilistic problem, intuition suggests that $F(k)$ peaks near $k$, with $\frac{n}{n+m} = \frac{k}{s}$. Indeed, in the limit case $s = n+m$, $k$ had better be $n$ for the value to survive. However, I'm not satisfied with the limit case.

Given that $F$ does peak around $k = k^* := \frac{ns}{n+m}$, I want to know how much it peaks. Thus the following question:

Find the smallest possible non-negative integer $\epsilon$ such that

$$ \sum_{k \in [k^*-\epsilon, k^*+\epsilon]} F(k) > 90\%.$$

The answer $\epsilon$ depends on $n, m, s$, and "$90\%$".

Answer 1

$\newcommand\ep{\epsilon}$

Of course, there is no simple explicit expression for the smallest $\ep$. However, one can give upper bounds on or approximations of this $\ep$. Indeed, $F$ is the probability mass function of a random variable, say $X$, with the hypergeometric distribution with parameters $n,m,s$. So, by a Hoeffding inequality, for $\ep\in[0,k^*]$ $$S(\ep):=\sum_{k\in[k^*-\ep,k^*+\ep]}F(k)=P(|X-k^*|\le\ep) \ge1-2e^{-2\ep^2/s}.$$ Solving now the equation $1-2e^{-2\ep^2/s}=0.90$ for $\ep>0$, we get a desired upper bound on the smallest $\ep$ such that $S(\ep)\ge0.90$.

If $s$ is much smaller than $\min(n,m)$ and if $\frac n{n+m}$ is not close to $0$ or $1$, then one can use a normal approximation to hypergeometric distribution.