Wenbo V. Li

Gaussian processes: inequalities, small ball probabilities and applications

Publisher Summary This chapter focuses on the inequalities, small ball probabilities, and application of Gaussian processes. It is well-known that the large deviation result plays a fundamental role in studying the upper limits of Gaussian processes, such as the Strassen type law of the iterated logarithm. However, the complexity of the small ball estimate is well-known, and there are only a few Gaussian measures for which the small ball probability can be determined completely. The small ball probability is a key step in studying the lower limits of the Gaussian process. It has been found that the small ball estimate has close connections with various approximation quantities of compact sets and operators, and has a variety of applications in studies of Hausdorff dimensions, rate of convergence in Strassens law of the iterated logarithm, and empirical processes.

The Gaussian measure of shifted balls

SummaryLet μ be a centered Gaussian measure on a Hilbert spaceH and let

Small Ball Estimates for Gaussian Processes under Sobolev Type Norms

B_R \subseteq H

On the future infima of some transient processes

be the centered ball of radiusR>0. Fora∈H and

Capture time of Brownian pursuits

\mathop {\lim }\limits_{t{\mathbf{ }} \to {\mathbf{ }}\infty } {\mathbf{ }}R(t)/t< {\mathbf{ }}||a||

A CLT for the L2 modulus of continuity of Brownian local time

, we give the exact asymptotics of μ(BR(t)+t·a) ast→∞. Also, upper and lower bounds are given when μ is defined on an arbitrary separable Banach space. Our results range from small deviation estimates to large deviation estimates.