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Decomposition and Limit Theorems for a Class of Self-Similar Gaussian Processes

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Stochastic Analysis and Related Topics

Part of the book series: Progress in Probability ((PRPR,volume 72))

Abstract

We introduce a new class of self-similar Gaussian stochastic processes, where the covariance is defined in terms of a fractional Brownian motion and another Gaussian process. A special case is the solution in time to the fractional-colored stochastic heat equation described in Tudor (Analysis of variations for self-similar processes: a stochastic calculus approach. Springer, Berlin, 2013). We prove that the process can be decomposed into a fractional Brownian motion (with a different parameter than the one that defines the covariance), and a Gaussian process first described in Lei and Nualart (Stat Probab Lett 79:619–624, 2009). The component processes can be expressed as stochastic integrals with respect to the Brownian sheet. We then prove a central limit theorem about the Hermite variations of the process.

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References

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Acknowledgements

D. Nualart is supported by NSF grant DMS1512891 and the ARO grant FED0070445

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Correspondence to David Nualart .

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Harnett, D., Nualart, D. (2017). Decomposition and Limit Theorems for a Class of Self-Similar Gaussian Processes. In: Baudoin, F., Peterson, J. (eds) Stochastic Analysis and Related Topics. Progress in Probability, vol 72. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59671-6_5

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