The Annals of Probability

Large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes

Xia Chen, Wenbo V. Li, Jan Rosiński, and Qi-Man Shao

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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.

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Ann. Probab., Volume 39, Number 2 (2011), 729-778.

First available in Project Euclid: 25 February 2011

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Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion 60J55: Local time and additive functionals 60F10: Large deviations 60G15: Gaussian processes 60G18: Self-similar processes

Local time intersection local time large deviations fractional Brownian motion Riemann–Liouville process law of iterated logarithm


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. 39 (2011), no. 2, 729--778. doi:10.1214/10-AOP566.

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