Fractional Brownian motions as "higher-order"
fractional derivatives of Brownian local times
Endre Csáki, Zhan Shi and Marc Yor
Summary: Fractional derivatives ${\cal D}^\gamma$ of Brownian
local times are well defined for all $\gamma<3/2$. We show that, in the
weak convergence sense, these fractional derivatives admit themselves
derivatives which feature all fractional Brownian motions. Strong
approximation results are also developed as counterparts of limit
theorems for Brownian additive functionals which feature the fractional
derivatives of Brownian local times.
Keywords: Local time, additive functional, principal value,
Brownian sheet, fractional Brownian motion, Hilbert transform,
fractional derivative.
AMS 1991 subject classification: 60J55, 60J65, 60F05.