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 Electronic Journal of Probability > Vol. 8 (2003) > Paper 13 open journal systems 


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.


Full text: PDF

Pages: 1-20

Published on: August 19, 2003
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Electronic Journal of Probability. ISSN: 1083-6489