Let $P$ be a probability distribution on a finite set $\mathcal{X}$ and let $X_1, X_2, \ldots, X_n$ be drawn i.i.d. according to $P$. Define the empirical distribution:

$\hat{P_n}(x) = \frac{1}{n} \sum_{i=1}^{n} 1_{X_i = x}$

Let $d_H(P,Q)$ be the Hellinger distance:

$d_H(P,Q) = \left( \frac{1}{2} \sum_{x \in \mathcal{X}} ( \sqrt{P(x)} - \sqrt{Q(x)} )^2 \right)^{1/2}$

Is there a nice expression for the expected distance between $\hat{P_n}$ and $P$? That is, is there some formula like

$\mathbb{E}[ d_H(P,Q) ] = C \frac{1}{n} - O(\frac{1}{n^2})$

where $C$ can be written out explicitly? Or if the rate of convergence is slower than $1/n$, can we get the exact rate of convergence?

For context, if we consider the KL-divergence or $L_1$ distance then we can get explicit expressions for the first term in the rate of convergence of $\hat{P_n}$ to $P$. Can we do the same for the Hellinger distance?

It would be interesting to know this for densities as well, but maybe the discrete problem is easier.