La semaine de l’IMB

The Ewens-Pitman model is a distribution on random partitions of {1, . . . , n} indexed by two parameters $\alpha\in$ [0,1) and $\theta$ > - $\alpha$. We establish, for large sample size n, a law of large numbers, a central limit theorem for the number of blocks and a joint central limit theorem for the numbers of blocks of given size in the Ewens-Pitman random partition under the ''large $\theta$'' regime, where the parameter $\theta$ depends linearly on n. We conclude by showing two statistical applications of our results: a posterior version of the CLT for the number of blocks is used to perform uncertainty quantification for the Bayesian  nonparametrics approach to species sampling problems, while the joint CLT is used to develop a novel estimator for the parameters of the model. 

This talk is based on (partly ongoing) joint work with B. Bercu, E. Dolera and S. Favaro.

In this talk, we are interested in the group Aut(X,0) of automorphisms of the germ (X,0) of a complex analytic space. We show that Aut(X,0) contains a contracting automorphism if and only if the singularity (X,0) is quasi-homogeneous. This extends previous results by Favre and Ruggiero for the normal surface case, as well as the recent results by Ornea and Verbitsky for the isolated singularity case. Our proof relies on embeddings and normal forms techniques.