Jan Rosinski
Mathematics/UTK

PREPRINTS:


Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes (with Xia Chen, Wenbo V. Li, and Qi-Man Shao)

ABSTRACT. 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.

Inverse problems for regular variation of linear filters, a cancellation property for sigma-finite measures, and identification of stable laws (with Martin Jacobsen, Thomas Mikosch, and Gennady Samorodnitsky)

ABSTRACT. We study a group of related problems: the extent to which the presence of regular variation in the tail of certain sigma-finite measures at the output of a linear filter determines the corresponding regular variation of a measure at the input to the filter. This turns out to be related to the presence of a particular cancellation property in sigma-finite measures, which, in turn, is related to the uniqueness of the solution of certain functional equations. The techniques we develop are applied to weighted sums of iid random variables, to products of independent random variables, and to stochastic integrals with respect to Levy motions.
(To appear in the Annals of Applied Probability.)

General Upsilon-transformations (with Ole Barndorff-Nielsen and Steen Thorbjornsen)

ABSTRACT. In this paper we introduce a general class of transformations of (all or most of) the class of d-dimensional Levy measures on Rd, into itself. We refer to transformations of this type as Upsilon-transformations. Closely associated to these are mappings of the set of all infinitely divisible laws on Rd into itself. In considerable generality, the mappings are one-to-one, regularising and bi-continuous. Furthermore, in many cases the transformations have a stochastic interpretation in terms of stochastic integrals with respect to Levy processes.
(To appear in the ALEA - Latin American Journal Of Probability And Mathematical Statistics.)

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Last Modified: February, 2008.