Open Access
August 2016 Functional limit theorems for generalized variations of the fractional Brownian sheet
Mikko S. Pakkanen, Anthony Réveillac
Bernoulli 22(3): 1671-1708 (August 2016). DOI: 10.3150/15-BEJ707


We prove functional central and non-central limit theorems for generalized variations of the anisotropic $d$-parameter fractional Brownian sheet (fBs) for any natural number $d$. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and on the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application, we derive functional limit theorems for power variations of the fBs and discuss what is a proper way to interpolate them to ensure functional convergence.


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Mikko S. Pakkanen. Anthony Réveillac. "Functional limit theorems for generalized variations of the fractional Brownian sheet." Bernoulli 22 (3) 1671 - 1708, August 2016.


Received: 1 August 2014; Revised: 1 January 2015; Published: August 2016
First available in Project Euclid: 16 March 2016

zbMATH: 1338.60099
MathSciNet: MR3474829
Digital Object Identifier: 10.3150/15-BEJ707

Keywords: central limit theorem , Fractional Brownian sheet , Hermite sheet , Malliavin calculus , Non-central limit theorem , power variation

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 3 • August 2016
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