In this paper, we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes. We also show that a fractional Brownian motion and the related Riemann–Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional Brownian motion; (2) logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of Riemann–Liouville processes; (3) comparison of local and intersection local times based on embedding of a part of a fractional Brownian motion into the reproducing kernel Hilbert space of the Riemann–Liouville process.
Publié le : 2011-03-15
Classification:
Local time,
intersection local time,
large deviations,
fractional Brownian motion,
Riemann–Liouville process,
law of iterated logarithm,
60G22,
60J55,
60F10,
60G15,
60G18
@article{1298669178,
author = {Chen, Xia and Li, Wenbo V. and Rosi\'nski, Jan and Shao, Qi-Man},
title = {Large deviations for local times and intersection local times of fractional Brownian motions and Riemann--Liouville processes},
journal = {Ann. Probab.},
volume = {39},
number = {1},
year = {2011},
pages = { 729-778},
language = {en},
url = {http://dml.mathdoc.fr/item/1298669178}
}
Chen, Xia; Li, Wenbo V.; Rosiński, Jan; Shao, Qi-Man. Large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes. Ann. Probab., Tome 39 (2011) no. 1, pp. 729-778. http://gdmltest.u-ga.fr/item/1298669178/