Second moment of a mixture likelihood ratio (${\chi }^2({\sum }_j{w}_j{P}_j\Vert Q)={\sum }_{j,k}{w}_j{w}_k{\mathbb{E}}_Q[L_j{L}_k]-1$)

If $L_j=d{P}_j/dQ$ and the nonnegative mixture weights sum to one, the likelihood ratio of the mixture model is $L={\sum }_j{w}_j{L}_j$. Expanding ${\mathbb{E}}_Q{L}^2-1$ proves the displayed chi-squared divergence identity. For independent random variables sampled from a fixed component, one-observation cross moments multiply across observations. The component is drawn once for the whole sample; redrawing it per observation gives a different mixture model.