Laplace Transforms via Hadamard Factorization
Fuchang Gao, University of Idaho
Jan Hannig, Colorado State University
Tzong-Yow Lee, University of Maryland
Fred Torcaso, The Johns Hopkins University
Abstract
In this paper we consider the Laplace transforms of
some random series, in particular, the random series derived as
the squared $L_2$ norm of a Gaussian stochastic process.
Except for some special cases,
closed form expressions for Laplace transforms
are, in general, rarely obtained. It is
the purpose of this paper to show that for many Gaussian random
processes the Laplace transform can be expressed in terms of well
understood functions using complex-analytic theorems on infinite
products, in particular, the Hadamard Factorization Theorem.
Together with some tools from linear differential operators, we
show that in many cases the Laplace transforms can be obtained
with little work. We demonstrate this on several examples.
Of course, once the Laplace transform is known explicitly one can
easily calculate the corresponding
exact $L_2$ small ball probabilities using Sytaja
Tauberian theorem. Some generalizations are mentioned.
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