= Chi-squared divergence of product measures
{title2=$1+\chi^2(Q^{\otimes n}\Vert P^{\otimes n})=(1+\chi^2(Q\Vert P))^n$}
If $r=dQ/dP$, <independence> gives $E_{P^{\otimes n}}\prod_i r(X_i)^2=(E_Pr^2)^n$. Since $E_Pr=1$, this proves the displayed identity. The <Cauchy-Schwarz inequality> also gives $\|Q^{\otimes n}-P^{\otimes n}\|_1^2\leq\chi^2(Q^{\otimes n}\Vert P^{\otimes n})$. 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>.
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