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


$$
D\_{\text{Hell}}(P\|\hat{P}\_N) := \left(\sum\_{i=1}^k\left(\sqrt{p\_i}-\sqrt{N\_i/N}\right)^2 \right)^{1/2}=2\left(1-\sum\_{i=1}^k\sqrt{p\_i}\sqrt{N\_i/N}\right)^{1/2}.
$$


Using the Markov inequality, [(Matusita 1995)](https://projecteuclid.org/download/pdf_1/euclid.aoms/1177728422) showed the **non-asymptotic** bound



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


The author also proved the convergence in law



> 
> **Theorem II.**
> $$4ND\_{\text{Hellinger}}(P\|\hat{P}\_N)^2 \overset{\mathcal L}{\longrightarrow}\chi\_{(k-1)}^2.
> $$
> 
> 
> 


Question
========


Can the **non-asymptotic** tail bound in Theorem I above be improved ?
Maybe something like the [sub-exponential tail bound for the chi-squared distribution](https://projecteuclid.org/download/pdf_1/euclid.aos/1015957395).



Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$
This is what I've come up with. It's too long to be a comment, so I decided to post it as an answer.


So, it was proven in *LeCam, L. M. (1969). Théorie Asymptotique de la Décision Statistique, p35* that $D\_{\text{Hell}}(\cdot\|\cdot)^2/2 \le TV(\cdot, \cdot) \le D\_{\text{Hell}}(\cdot\|\cdot)$. On the other hand, [Theorem 2 of this paper](https://arxiv.org/abs/1205.6711) proves a powerful **nonasymptotic** tail bound on the $TV(P\|\hat{P}\_N)$, namely



> 
> For every $\epsilon \ge \sqrt{k/N}$, one has
>  $$
> P(TV(P\|\hat{P}\_N) > \epsilon) \le \exp\left(-\frac{N}{2}(\epsilon-\sqrt{k/N})^2\right).
> $$
> 
> 
> 


Putting things together, we have



> 
> For every $\epsilon \ge 2\sqrt{k/N}$, one has
>  $$
> P(D\_{\text{Hell}}(P\|\hat{P}\_N)^2 > \epsilon) \le \exp\left(-\frac{N}{2}(\epsilon/2-\sqrt{k/N})^2\right).
> $$
> 
> 
> 


from which one may immediate recover a **nonasymptotic** sub-exponential tail bound for $D\_{\text{Hell}}(P\|\hat{P}\_N)$, as dreamed, for example



> 
> For every $\epsilon \ge 4\sqrt{5k/N}$, one has
>  $$
> P(D\_{\text{Hell}}(P\|\hat{P}\_N) > \sqrt{\epsilon}) \le \exp(-0.075N\epsilon^2).
> $$
> 
> 
> 


