# Small Value Probability and Metric Entropy

Friday, September 30, 2011 - 11:00am - 12:00pm

Keller 3-180

Wenbo Li (University of Delaware)

Small value probabilities or small deviations study the decay probability that positive random variables behave near zero.

In particular, small ball probabilities provide the asymptotic behavior of the probability measure inside a ball as the radius of the

ball tends to zero.

Metric entropy is defined as the logarithmic of the minimum covering number of compact set by balls of very small radius.

In this talk, we will provide an overview on precise connections between the small value probability

of Gaussian process/measure and the metric entropy of the associated compact operator.

Interplays and applications to many related problems/areas will be given, along with various fundamental tools and techniques from high dimensional probability theory.

Throughout, we use Brownian motion (integral operator) and Brownian

sheets (tensored Brownian motion and operator) as illustrating examples.

In particular, small ball probabilities provide the asymptotic behavior of the probability measure inside a ball as the radius of the

ball tends to zero.

Metric entropy is defined as the logarithmic of the minimum covering number of compact set by balls of very small radius.

In this talk, we will provide an overview on precise connections between the small value probability

of Gaussian process/measure and the metric entropy of the associated compact operator.

Interplays and applications to many related problems/areas will be given, along with various fundamental tools and techniques from high dimensional probability theory.

Throughout, we use Brownian motion (integral operator) and Brownian

sheets (tensored Brownian motion and operator) as illustrating examples.