Bernoulli

Integral representations and properties of operator fractional Brownian motions

Gustavo Didier and Vladas Pipiras

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Abstract

Operator fractional Brownian motions (OFBMs) are (i) Gaussian, (ii) operator self-similar and (iii) stationary increment processes. They are the natural multivariate generalizations of the well-studied fractional Brownian motions. Because of the possible lack of time-reversibility, the defining properties (i)–(iii) do not, in general, characterize the covariance structure of OFBMs. To circumvent this problem, the class of OFBMs is characterized here by means of their integral representations in the spectral and time domains. For the spectral domain representations, this involves showing how the operator self-similarity shapes the spectral density in the general representation of stationary increment processes. The time domain representations are derived by using primary matrix functions and taking the Fourier transforms of the deterministic spectral domain kernels. Necessary and sufficient conditions for OFBMs to be time-reversible are established in terms of their spectral and time domain representations. It is also shown that the spectral density of the stationary increments of an OFBM has a rigid structure, here called the dichotomy principle. The notion of operator Brownian motions is also explored.

Article information

Source
Bernoulli, Volume 17, Number 1 (2011), 1-33.

Dates
First available in Project Euclid: 8 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1297173831

Digital Object Identifier
doi:10.3150/10-BEJ259

Mathematical Reviews number (MathSciNet)
MR2797980

Zentralblatt MATH identifier
1284.60079

Keywords
dichotomy principle integral representations long-range dependence multivariate Brownian motion operator fractional Brownian motion operator self-similarity time-reversibility

Citation

Didier, Gustavo; Pipiras, Vladas. Integral representations and properties of operator fractional Brownian motions. Bernoulli 17 (2011), no. 1, 1--33. doi:10.3150/10-BEJ259. https://projecteuclid.org/euclid.bj/1297173831


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