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Fredholm representation of multiparameter Gaussian processes with applications to equivalence in law and series expansions✩
Volume 2, Issue 3 (2015): PRESTO-2015, pp. 287–295
Tommi Sottinen   Lauri Viitasaari 1  

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https://doi.org/10.15559/15-VMSTA39CNF
Pub. online: 2 October 2015      Type: Research Article      Open accessOpen Access

✩ The authors thank the referees for their useful comments.
1 Lauri Viitasaari was partially funded by Emil Aaltonen Foundation.

Received
25 June 2015
Revised
21 September 2015
Accepted
21 September 2015
Published
2 October 2015

Abstract

We show that every multiparameter Gaussian process with integrable variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square root of the covariance. We analyze the equivalence of multiparameter Gaussian processes by using the Fredholm representation and show how to construct series expansions for multiparameter Gaussian processes by using the Fredholm kernel.

References

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Berlinet, A., Thomas-Agnan, C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer Academic Publishers, Boston, MA (2004), 355 p. MR2239907. doi:10.1007/978-1-4419-9096-9
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Sottinen, T., Tudor, C.A.: On the equivalence of multiparameter Gaussian processes. J. Theor. Probab. 19(2), 461–485 (2006). MR2283386 (2008e:60100). doi:10.1007/s10959-006-0022-5
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Sottinen, T., Viitasaari, L.: Stochastic analysis of Gaussian processes via Fredholm representation. Preprint, arXiv:1410.2230 (2014)

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Keywords
Equivalence in law Gaussian sheets multiparameter Gaussian processes representation of Gaussian processes series expansions

MSC2010
60G15 60G60

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