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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 63, Pages 3423-3434
http://dx.doi.org/10.1155/S0161171204210183

On the asymptotic behavior of the second moment of the Fourier transform of a random measure

1Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa (FCT/UNL), Quinta da Torre, Caparica 2829-516, Portugal
2Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa (CMAF/UL), Lisboa 1649-003, Portugal

Received 21 October 2002

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The behavior at infinity of the Fourier transform of the random measures that appear in the theory of multiplicative chaos of Mandelbrot, Peyrière, and Kahane is an area quite unexplored. For context and further reference, we first present an overview of this theory and then the result, which is the main objective of this work, generalizing a result previously announced by Kahane. We establish an estimate for the asymptotic behavior of the second moment of the Fourier transform of the limit random measure in the theory of multiplicative chaos. After looking at the behavior at infinity of the Fourier transform of some remarkable functions and measures, we prove a formula essentially due to Frostman, involving the Riesz kernels.