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We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, $d_H^2(P,Q) = \frac{1}{{2}}\left|\left| \sqrt{P}-\sqrt{Q}\right|\right|_2^2$. That is:

\begin{equation} d_H^2(P,Q)\leq d_{TV}(P,Q). \end{equation}

I noticed, however, that when $P$ and $Q$ are Gaussian, ${\cal N}(-\mu,1)$ and ${\cal N}(\mu,1)$, $\mu\in \mathbb{R}$, truncated in the interval $[a,b]$, $a,b\in \mathbb{R}$, the following tighter bound seems to hold:

\begin{equation} d_H^2(P,Q)\leq d_H(P,Q)\leq d_{TV}(P,Q). \end{equation}

I have repeatedly verified the above inequality with numerical simulations, but I could not prove that it holds for all parameters $a,b$ and $\mu$.

If you have any pointers or ideas on how to approach this problem, I would really appreciate your input!

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