Three-Toed Sloth

"The Universal Glivenko-Cantelli Property" (Next Week at the Statistics Seminar)

For the first statistics seminar of 2011, we are very happy to welcome —

Ramon van Handel, "The Universal Glivenko-Cantelli Property"

Abstract: Uniform laws of large numbers are basic tools in many problems in probability theory, statistics, and machine learning. On the other hand, the law of large numbers is ultimately "just" a special case of two fundamental probabilistic limit theorems: the reverse martingale convergence theorem and the pointwise ergodic theorem. What can one say about uniform convergence in the more general setting? Surprisingly, it turns out that for a given class of functions, universal uniform convergence in the law of large numbers, the reverse martingale convergence theorem, and the pointwise ergodic theorem are all equivalent. Moreover, such classes of functions (which are more general than the well-known Vapnik-Chervonenkis classes) can be characterized by certain geometric and combinatorial properties. As an application, I will discuss the pathwise optimality of sequential decisions under partial information.

Place and time: Scaife Hall 125, on Monday, 24 January 2011, 4--5 pm

As always, the seminar is free and open to the public.

Enigmas of Chance