Évènements

In this talk, we consider the problem of testing for the sphericity of a collection of random vectors. It is well known that in the classical elliptical model, testing for rotational symmetry of the underlying distribution is equivalent to testing that a scatter parameter is a multiple of the identity matrix. We consider the more general model of random vectors with elliptical directions introduced by R.H. Randles and present a few scenarios where testing for sphericity is still equivalent to testing that the scatter parameter is a multiple of the identity. These new scenarios include, for instance, non-classical settings where some dependence of a rather general form studied here for the first time may be present between observations. We study, under these new assumptions, the behavior of the classical spatial sign test and show that under certain mild assumptions, the test is asymptotically valid and has the same local asymptotic power as in the classical elliptical scenario. We then show that, contrary to some commonly held belief, the spatial sign test enjoys some local asymptotic optimality properties when it comes to testing for sphericity when the underlying distribution is strongly heavy-tailed.

(L’exposé sera en français, avec des slides en anglais.)

Random multiplicative functions are random models for arithmetic functions such as Dirichlet characters. Moments of sums involving random multiplicative functions are related to interesting counting problems, and understanding these counts can allow one to deduce the limiting distribution of the sums. Using this idea, Benatar, Nishry, and Rodgers showed that the limiting distribution of exponential sums with random multiplicative coefficients is Gaussian. However, they found that moments do not suffice if one wishes to understand the maximum size of these exponential sums. After introducing random multiplicative functions, we will discuss why this is the case, and show how one can obtain conjecturally sharp lower bounds for the maximum size of exponential sums with random multiplicative coefficients.