About the asymptotic accuracy of Barron density estimates

By extending the information-theoretic arguments of previous papers dealing with the Barron-type density estimates, and their consistency in information divergence and chi-square divergence, the problem of consistency in Csiszar´s φ-divergence is motivated for general convex functions φ. The problem of consistency in φ-divergence is solved for all φ with φ(0)<∞ and φ(t)=O(t ln t) when t→∞. The problem of consistency in the expected φ-divergence is solved for all φ with tφ(1/t)+φ(t)=O(t²) when t→∞. Various stronger versions of these asymptotic restrictions are considered too. Assumptions about the model needed for the consistency are shown to depend on how strong these restrictions are