Jan Rosinski
Mathematics/UTK
PREPRINTS:
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.
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.)
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.