A density is approximable in KL divergence by finite GMMs only if it has finite second moment, with sufficiency holding for finite log-moment positive densities and countable-scale support-aware classes via pointwise likelihood ratio convergence and uniform integrability.
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Characterisations of Kullback--Leibler approximation by finite Gaussian mixtures
A density is approximable in KL divergence by finite GMMs only if it has finite second moment, with sufficiency holding for finite log-moment positive densities and countable-scale support-aware classes via pointwise likelihood ratio convergence and uniform integrability.