Electronic Journal of Probability

Nonexistence of fractional Brownian fields indexed by cylinders

Nil Venet

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We show in this article that there exists no $H$-fractional Brownian field indexed by the cylinder $\mathbb{S} ^{1} \times ]0,\varepsilon [$ endowed with its product distance $d$ for any $\varepsilon >0$ and $H>0$. This is equivalent to say that $d^{2H}$ is not a negative definite kernel, which also leaves us without a proof that many classical stationary kernels, such that the Gaussian and exponential kernels, are positive definite kernels – or valid covariances – on the cylinder.

We generalise this result from the cylinder to any Riemannian Cartesian product with a minimal closed geodesic. We also investigate the case of the cylinder endowed with a distance asymptotically close to the product distance in the neighbourhood of a circle.

As a consequence of our result, we show that the set of $H$ such that $d^{2H}$ is negative definite behaves in a discontinuous way with respect to the Gromov-Hausdorff convergence on compact metric spaces.

These results extend our comprehension of kernel construction on metric spaces, and in particular call for alternatives to classical kernels to allow for Gaussian modelling and kernel method learning on cylinders.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 75, 26 pp.

Received: 1 June 2017
Accepted: 7 December 2018
First available in Project Euclid: 3 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion 60G60: Random fields

positive definite kernel valid covariance fractional brownian motion random field Riemannian manifold

Creative Commons Attribution 4.0 International License.


Venet, Nil. Nonexistence of fractional Brownian fields indexed by cylinders. Electron. J. Probab. 24 (2019), paper no. 75, 26 pp. doi:10.1214/18-EJP256. https://projecteuclid.org/euclid.ejp/1562119475

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