A new bound on the cumulant generating function of Dirichlet processes

Pierre Perrault, Denis Belomestny, Pierre Ménard, Éric Moulines, Alexey Naumov, Daniil Tiapkin, Michal Valko

Preprint

Abstract

We introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP), using superadditivity. Our key technical contribution is the demonstration of the superadditivity of the log-moment generating function of the DP. This result, combined with Fekete's lemma and Varadhan's integral lemma, converts the known asymptotic large deviation principle into a practical upper bound on the CGF for any scale parameter. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence. This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.

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