Characteristic functionals of Dirichlet measures

Lorenzo Dello Schiavo

doi:10.1214/19-ejp371

Abstract

We compute characteristic functionals of Dirichlet-Ferguson measures over a locally compact Polish space and prove continuous dependence of the random measure on the parameter measure. In finite dimension, we identify the dynamical symmetry algebra of the characteristic functional of the Dirichlet distribution with a simple Lie algebra of type $A$. We study the lattice determined by characteristic functionals of categorical Dirichlet posteriors, showing that it has a natural structure of weight Lie algebra module and providing a probabilistic interpretation. A partial generalization to the case of the Dirichlet-Ferguson measure is also obtained.

Highlights

Introduction and main resultsLet X be a locally compact Polish space with Borel σ-algebra B(X) and let P(X) be the space of probability measures on (X, B(X))
For σ ∈ P(X) we denote by Dσ the Dirichlet–Ferguson measure [9] on P(X) with probability intensity σ
This led to the introduction of different characterizing transforms, inversion formulas based on characteristic functionals of other random measures, and, at least in the case X = R, to the celebrated Markov–Krein identity

Summary

Universitat Bonn

We compute characteristic functionals of Dirichlet–Ferguson measures over a locally compact Polish space and prove continuous dependence of the random measure on the parameter measure. Whereas Theorem 1.1 allows for Bochner–Minlos and Levy Continuity related results to come into play, the non-multiplicativity of Dσ (corresponding to the non-infinite-divisibility of the measure) immediately rules out the usual approach to quasi-invariance via Fourier transforms [2, 20, 42, 43]. Other approaches to this problem rely on finite-dimensional approximation techniques, variously concerned with approximating the space [34, 35], the σ-algebra [20] or the acting group [11, 45]. By the classical theory of characteristic functionals we recover known asymptotic expressions for Dβσ when β → 0, ∞ is a real parameter (Cor. 3.13, cf. [37, p. 311]), propose a Gibbsean interpretation thereof (Rem. 3.14), and prove analogous expressions for the entropic measure β

Pσ on compact

Since every finite measure on a Polish space is