Fourier transform of general stochastic measures
Authors:
V. M. Radchenko and N. O. Stefans’ka
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 94 (2017), 151-158
MSC (2010):
Primary 60G57, 60H15, 60H05
DOI:
https://doi.org/10.1090/tpms/1015
Published electronically:
August 25, 2017
MathSciNet review:
3553460
Full-text PDF
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Additional Information
Abstract: The Fourier transform is defined for general stochastic measures in $\mathbb {R}^d$. The inversion theorem for this transform is proved and a connection to the convergence of stochastic integrals is established. An example of applications of this result is considered for the convergence of solutions of the stochastic heat equation.
References
- Stanisław Kwapień and Wojbor A. Woyczyński, Random series and stochastic integrals: single and multiple, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1167198
- V. N. Radchenko, Integrals of some random functions with respect to general random measures, Ukraïn. Mat. Zh. 51 (1999), no. 8, 1087–1095 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 51 (1999), no. 8, 1226–1236 (2000). MR 1719308, DOI https://doi.org/10.1007/BF02592510
- V. Radchenko, Riemann integral of a random function and the parabolic equation with a general stochastic measure, Theory Probab. Math. Statist. 87 (2013), 185–198. Translation of Teor. Ǐmovīr. Mat. Stat. No. 87 (2012), 163–175. MR 3241455, DOI https://doi.org/10.1090/S0094-9000-2014-00912-6
- Gennady Samorodnitsky and Murad S. Taqqu, Stable non-Gaussian random processes, Stochastic Modeling, Chapman & Hall, New York, 1994. Stochastic models with infinite variance. MR 1280932
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
- Vadym Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554, DOI https://doi.org/10.4064/sm194-3-2
References
- S. Kwapień and W. A. Woycziński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992. MR 1167198
- V. N. Radchenko, Integrals with respect to general random measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine 27 (1999). (Russian) MR 1719308
- V. Radchenko, Riemann integral of a random function and the parabolic equation with a general stochastic measure, Teor. Imovirnost. Matem. Statist. 87 (2012), 163–175; English transl. in Theor. Probability and Math. Statist. 87 (2013), 185–198. MR 3241455
- G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, Boca Raton, 1994. MR 1280932
- E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, no. 32, Princeton University Press, Princeton, NJ, 1971. MR 0304972
- V. M. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554
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Additional Information
V. M. Radchenko
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
vradchenko@univ.kiev.ua
N. O. Stefans’ka
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
valentinasavych@mail.ru
Keywords:
Stochastic measure,
Fourier transform of stochastic processes,
weak convergence,
stochastic heat equation
Received by editor(s):
April 28, 2016
Published electronically:
August 25, 2017
Article copyright:
© Copyright 2017
American Mathematical Society
