Exercise 15.5: Le Cam's Inequality

chapter 15

Note that \begin{equation} \abs{p - q} = \abs{\sqrt{p} - \sqrt{q}}\abs{\sqrt{p} + \sqrt{q}}, \quad (\sqrt{p} + \sqrt{q})^2 = 2p + 2q - (\sqrt{p} - \sqrt{q})^2. \end{equation} Therefore, by Cauchy–Schwarz, \begin{equation} \TV^2(\P, \Q) \le \tfrac14 h^2(\P, \Q)(4 - h^2(\P, \Q)). \end{equation}