Small Ball Probabilities, Metric Entropy and Gaussian Rough Paths

William Salkeld

Published 2020-02-15Version 1

We study the Small Ball Probabilities of Gaussian Rough Paths. While many works on Rough Paths study the Large Deviations Principles for Stochastic Processes driven by Gaussian Rough paths, it is a noticeable gap in the literature that Small Ball Probabilities have not been extended to the Rough Path framework. LDPs provide macroscopic information about a measure for establishing Integrability type properties. Small Ball Probabilities provide microscopic information and are used to establish a locally accurate approximation for a random variable. Given the compactness of a Reproducing Kernel Hilbert space Ball, its Metric Entropy provides invaluable information on how to approximate the law of a Gaussian Rough path. As an application, we are able to find upper and lower bounds for the rate of convergence of an Empirical Rough Gaussian measure to its true law in pathspace.