## Electronic Journal of Probability

### Laplace Transforms via Hadamard Factorization

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 8 (2003), paper no. 13, 20 p.

Dates
First available in Project Euclid: 23 May 2016

https://projecteuclid.org/euclid.ejp/1464037586

Digital Object Identifier
doi:10.1214/EJP.v8-151

Mathematical Reviews number (MathSciNet)
MR1998764

Zentralblatt MATH identifier
1064.60061

Subjects
Primary: 60G15: Gaussian processes

#### Citation

Gao, Fuchang; Hannig, Jan; Lee, Tzong-Yow; Torcaso, Fred. Laplace Transforms via Hadamard Factorization. Electron. J. Probab. 8 (2003), paper no. 13, 20 p. doi:10.1214/EJP.v8-151. https://projecteuclid.org/euclid.ejp/1464037586

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