I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant literature.
Consider a random variable $X$ distributed hypergeometrically with parameters $(n,m,i)$, i.e.,
$$p_X(x)=\frac{{i \choose x}{n-i \choose m-x}}{{n \choose m}},$$ where $\max(0,m+i-n)\leq x \leq \min(i,m)$.
The simple question I am asking is are there any good lower bounds for $$\sum_x (p_X(x))^2$$ as a function of $n$ specifically (a bound which holds for a given $n$ and which is not dependent on $i,m$)? Note: the summation is over the entire support of $X$, i.e., $\max(0,m+i-n)\leq x \leq \min(i,m)$.
My attempt and observations:
Using tail bounds (see 1) for the hypergeometric distribution gives $$\sum_{x:|x-\mathbb E X|\leq\sqrt n}p_X(x)\geq 1-2e^{-2}=c.$$ Now we can use Cauchy-Schwarz to get something along the lines of: \begin{align*} \sum_x p_X(x)^2 &\geq \sum_{x:|x-\mathbb E X|\leq\sqrt n}p_X(x)^2\\ &\geq \frac{1}{2\sqrt n} c^2=\frac{c'}{\sqrt n}. \end{align*}
This bound turns out to be weak for my purposes. I also computed the quantity for $n\in \{3,4,5,...,1000\}$ and for all relevant $i,m$ and it looks like $\sum_x p_X(x)^2 \geq \frac{1}{\log^2 n}$. This solves my problem if it is indeed true for all $n$.
Any help/leads would be appreciated. Thanks!