Approximation of fractional local times: Zero energy and derivatives

We consider empirical processes associated with high-frequency observations of a fractional Brownian motion (fBm) X Hurst parameter H?(0,1), and derive conditions under which these verify (possibly uniform) law large numbers, as well second order limit theorem. devote specific emphasis to the “zero energy” case, corresponding kernel whose integral on real line equals zero. Our asymptotic results are explicit rates convergence, expressed either in terms local time or its derivatives: particular, full force our finding applies “rough range” 0<H<1/3, previous literature has been mostly silent. The use derivatives times for studying fluctuations fBm is new, main technological breakthrough present paper. based Malliavin calculus Fourier analysis, extend complete several findings literature, example, by Jeganathan (Ann. Probab. 32 (2004) 1771–1795; (2006); (2008)) Podolskij Rosenbaum (J. Financ. Econom. 16 (2018) 588–598).