A New Bound on the Cumulant Generating Function of Dirichlet Processes

In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) X sim DP(αν_0), using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of αmapsto log E_{X sim DP(αν_0)}[exp( E_X[αf])], where E_X[f] = int f dX. 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 logE_{Xsim DP(αν_0)}{exp(E_{X}{[f]})} for any α> 0. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence αKL(ν_0Vert cdot). This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.