Large Deviations for the Dirichlet Processes

Abstract

The Dirichlet process and its two-parameter counterpart are random, purely atomic probabilities with masses distributed according to the Poisson–Dirichlet distribution and the two-parameter Poisson–Dirichlet distribution, respectively. The order by size of the masses does not matter, and the GEM distributions can then be used in place of the corresponding Poisson–Dirichlet distributions in the definition. When θ increases, these masses spread out evenly among the support of the type’s measure. Eventually the support is filled up and the types measure emerges as the deterministic limit. This resembles the behavior of empirical distributions for large samples. The focus of this chapter is the LDPs for the one- and two-parameter Dirichlet processes when θ tends to infinity. Relations of these results to Sanov’s theorem will be discussed. For simplicity, the space is chosen to be E=[0,1] in this chapter even though the results hold on any compact metric space. The diffuse probability ν ₀ on E will be fixed throughout the chapter.