Exercise 15.7: Decoupling for Hellinger Distance

chapter 15

Noting that \begin{equation} (\sqrt{p} - \sqrt{q})^2 = p + q - 2\sqrt{pq} \end{equation} we have the identity \begin{equation} \tfrac12 h^2(\P, \Q) = 1 - \int \sqrt{pq} \isd \mu, \end{equation} from which it immediately follows that \begin{equation} \tfrac12 h^2(\P_1 \otimes \cdots \otimes \P_n, \Q_1 \otimes \cdots \otimes \Q_n) = 1 - \prod_{i=1}^n(1 - \tfrac12 h^2(\P_i, \Q_i)). \end{equation}