Small ball properties and representation results

TL;DR

This paper develops new representation results for models with long memory by combining small ball probability estimates and Holder continuity assumptions, including estimates for Gaussian processes and fractional Brownian motion.

Contribution

It introduces novel small ball probability estimates for Gaussian processes with specific covariance structures and applies these to models with long memory.

Findings

Established small ball estimates for Gaussian processes with two-sided incremental variance bounds.

Derived small ball estimates for integral transforms of Wiener processes and fractional Brownian motion.

Provided new representation results for models exhibiting long memory effects.

Abstract

We show that small ball estimates together with Holder continuity assumption allow to obtain new representation results in models with long memory. In order to apply these results, we establish small ball probability estimates for Gaussian processes whose incremental variance admits two-sided estimates and the incremental covariance preserves sign. As a result, we obtain small ball estimates for integral transforms of Wiener processes and of fractional Brownian motion with Volterra kernels.