Concentration of the truncated variation of fractional Brownian motions of any Hurst index, their $1/H$-variations and local times

TL;DR

This paper derives optimal bounds for the deviation probabilities of truncated variation and $1/H$-variations of fractional Brownian motions for all Hurst indices, and proves convergence of crossing counts to local times.

Contribution

It provides the first tight deviation bounds for truncated variation and $1/H$-variations of fBm for any Hurst index, along with convergence results for crossing counts to local times.

Findings

Optimal deviation bounds for truncated variation of fBm.

Tight bounds for tails of $1/H$-variations of fBm.

Almost sure convergence of crossing counts to local times.

Abstract

We obtain bounds for probabilities of deviations of the truncated variation functional of fractional Brownian motions (fBm) of any Hurst index $H \in (0,1)$ from their expected values. Obtained bounds are optimal for large values of deviations up to multiplicative constants depending on the parameter $H$ only. As an application, we give tight bounds for tails of $1/H$-variations of fBm along Lebesgue partitions and establish the a.s. weak convergence (in $L^1$) of normalized numbers of strip crossings by the trajectories of fBm to their local times for any Hurst parameter $H \in (0,1)$.