Path regularity of Gaussian processes via small deviations

TL;DR

This paper investigates the almost sure regularity of Gaussian process sample paths by linking it to small deviation probabilities, establishing decay rates based on differentiability order.

Contribution

It provides a direct relation between the differentiability of Gaussian processes and the exponential decay rate of their small deviations, including non-integer differentiability.

Findings

Sample path regularity is connected to small deviation decay rates.

n-times differentiability implies a decay rate at most ε^{-1/n}.

Results extend to non-integer differentiability orders.

Abstract

We study the a.s. sample path regularity of Gaussian processes. To this end we relate the path regularity directly to the theory of small deviations. In particular, we show that if the process is $n$-times differentiable then the exponential rate of decay of its small deviations is at most $\varepsilon^{-1/n}$. We also show a similar result if $n$ is not an integer.