Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance between $P$ and $\hat{P}_N$, namely

$$
D_{\text{Hell}}(P\|\hat{P}_N) := \left(\sum_{i=1}k\left(\sqrt{p_i}-\sqrt{N_i/N}\right)^2 \right)^{1/2}=2\left(1-\sum_{i=1}^k\sqrt{pi}\sqrt{Ni/N}\right)^{1/2}.
$$
 
Using the Markov inequality, [(Matusita 1995)][1] showed **non-asymptotic** bound

> **Theorem I.** For all $t>0$, it holds that
$$
P(D_{\text{Hellinger}}(P\|\hat{P}_N)^2 \ge (k-1)t/N) \le 1/t.
$$

The author also proved the convergence convergence in law
> **Theorem II.**
$$4ND_{\text{Hellinger}}(P\|\hat{P}_N)^2 \overset{\mathcal L}{\longrightarrow}\chi_{(k-1)}^2.
$$

Combined with an [sub-exponential tail bound for the chi-squared distribution][2] and doing a bit of algebra, gives the **asymptotic** tail bound
> **Corollary.** For confidence level $\beta$ with every $0 < \beta \le exp(1-k) \le 1$, it holds that
$$
\liminf_{N\rightarrow \infty}P(D_{\text{Hellinger}}(P\|\hat{P}_N)^2 \ge 1.25\log(\beta^{-1})/N) \le \beta.
$$

Question
========
Can the **non-asymptotic** tail bound in Theorem I above be improved (ideally, to something close the the exponential bound in above Corollary) ?


  [1]: https://projecteuclid.org/download/pdf_1/euclid.aoms/1177728422
  [2]: https://projecteuclid.org/download/pdf_1/euclid.aos/1015957395