Decomposition and Limit Theorems for a Class of Self-Similar Gaussian Processes
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.
KeywordsFractional Brownian motion Hermite variations Self-similar processes Stochastic heat equation
AMS 2010 Classification60F05 60G18 60H07
D. Nualart is supported by NSF grant DMS1512891 and the ARO grant FED0070445
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