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2003 Laplace Transforms via Hadamard Factorization
Fuchang Gao, Jan Hannig, Tzong-Yow Lee, Fred Torcaso
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Electron. J. Probab. 8: 1-20 (2003). DOI: 10.1214/EJP.v8-151


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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Fuchang Gao. Jan Hannig. Tzong-Yow Lee. Fred Torcaso. "Laplace Transforms via Hadamard Factorization." Electron. J. Probab. 8 1 - 20, 2003.


Published: 2003
First available in Project Euclid: 23 May 2016

zbMATH: 1064.60061
MathSciNet: MR1998764
Digital Object Identifier: 10.1214/EJP.v8-151

Primary: 60G15

Keywords: Hadamard's factorization theorem , Laplace transforms , Small ball probability

Rights: Copyright © 2003 The Institute of Mathematical Statistics and the Bernoulli Society


Vol.8 • 2003
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