Bernoulli

  • Bernoulli
  • Volume 22, Number 3 (2016), 1671-1708.

Functional limit theorems for generalized variations of the fractional Brownian sheet

Mikko S. Pakkanen and Anthony Réveillac

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Abstract

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.

Article information

Source
Bernoulli, Volume 22, Number 3 (2016), 1671-1708.

Dates
Received: August 2014
Revised: January 2015
First available in Project Euclid: 16 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1458132995

Digital Object Identifier
doi:10.3150/15-BEJ707

Mathematical Reviews number (MathSciNet)
MR3474829

Zentralblatt MATH identifier
1338.60099

Keywords
central limit theorem fractional Brownian sheet Hermite sheet Malliavin calculus non-central limit theorem power variation

Citation

Pakkanen, Mikko S.; Réveillac, Anthony. Functional limit theorems for generalized variations of the fractional Brownian sheet. Bernoulli 22 (2016), no. 3, 1671--1708. doi:10.3150/15-BEJ707. https://projecteuclid.org/euclid.bj/1458132995


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