Uwe Einmahl
Vrije Universiteit Brussel
Title: Characterization of LIL behavior in Banach space.

In a previous joint work with Deli Li (Lakehead University, Thunder Bay, Canada) we have shown that the classical Hartman-Wintner LIL can be extended to a "law of a very slowly function''. Our proof of this result was based on some classical results for real-valued random variables. We now look into the corresponding problem in the general setting of Banach space valued random variables. Using some recent work on Bernstein type exponential inequalities we obtain a new exponential inequality for sums of independent B-valued random variables. This inequality immediately implies the infinite-dimensional version of the law of a very slowly varying function, but should have other applications as well. As a by-product of our work we can also report some progress on the problem of finding the precise value of the lim sup in the bounded Ledoux-Talagrand LIL (1988).

©2007, Department of Mathematical Sciences
Last Modified: April 11, 2007