Chi-squared divergence of product measures ($1+{\chi }^2(Q^{\otimes n}\Vert P^{\otimes n})=(1+{\chi }^2(Q\Vert P){)}^n$)

If $r=dQ/dP$, independence gives $E_{P^{\otimes n}}{\prod }_{i}r(X_i{)}^2=(E_P{r}^2{)}^n$. Since $E_{P}r=1$, this proves the displayed identity. The Cauchy-Schwarz inequality also gives $\Vert Q^{\otimes n}-P^{\otimes n}{\Vert }_1^2\le {\chi }^2(Q^{\otimes n}\Vert P^{\otimes n})$. Thus perturbations of size ${\chi }^2(Q\Vert P)=O(1/n)$ have a uniformly bounded joint total variation distance, a useful input to a metric squared-loss two-point bound.