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) showed the 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 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.

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) 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.

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) showed the 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 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 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) ?

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) showed the 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 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.

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) 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 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) ?

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) showed the 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 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 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) ?