Approximation of fractional local times: Zero energy and derivatives

Abstract

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