Concentration of the truncated variation of fractional Brownian motions of any Hurst index, their $1/H$-variations and 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 $H \in (0,1)$.