 Solution

For $P\ll Q$, the chi-squared divergence is ${\chi }^2(P\Vert Q)={\mathbb{E}}_Q[(dP/dQ-1{)}^2]$. Write $L_j=d{Q}_j/d{P}_0$ for the likelihood ratios of the complete $n$-ball sample. Since $d{P}_1/d{P}_0=d^{-1}{\sum }_j{L}_j$ and ${\mathbb{E}}_0{L}_j=1$, the second moment of a mixture likelihood ratio givesIf instead $L_j$ denotes a one-ball likelihood ratio, its cross moment must be raised to the $n$th power for the full sample, by independence.