Zuoqin Wang

On the Fundamental Group of Open Manifolds with Nonnegative Ricci Curvature

The authors establish some uniform estimates for the distance to halfway points of minimal geodesics in terms of the distantce to end points on some types of Riemannian manifolds, and then prove some theorems about the finite generation of fundamental group of Riemannian manifold with nonnegative Ricci curvature, which support the famous Milnor conjecture.

Differentiability of t-functionals of location and scatter

The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector μ and scatter matrix E of an elliptically symmetric t distribution on R d with degrees of freedom v > I extends to an M-functional defined on all probability distributions P in a weakly open, weakly dense domain U. Here U consists of P putting not too much mass in hyperplanes of dimension < d, as shown for empirical measures by Kent and Tyler [Ann. Statist. 19 (1991) 2102-2119]. It will be seen here that (μ, E) is analytic on U for the bounded Lipschitz norm, or for d = 1 for the sup norm on distribution functions. For k = 1, 2,..., and other norms, depending on k and more directly adapted to t functionals, one has continuous differentiability of order k, allowing the delta-method to be applied to (μ, Σ) for any P in U, which can be arbitrarily heavy-tailed. These results imply asymptotic normality of the corresponding M-estimators (μ n , Σ n In dimension d = 1 only, the t v functional (μ, a) extends to be defined and weakly continuous at all P.