Open Access
2019 Nonexistence of fractional Brownian fields indexed by cylinders
Nil Venet
Electron. J. Probab. 24: 1-26 (2019). DOI: 10.1214/18-EJP256


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.


Download Citation

Nil Venet. "Nonexistence of fractional Brownian fields indexed by cylinders." Electron. J. Probab. 24 1 - 26, 2019.


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

zbMATH: 07089013
MathSciNet: MR3978225
Digital Object Identifier: 10.1214/18-EJP256

Primary: 60G22 , 60G60

Keywords: fractional Brownian motion , Positive definite kernel , Random field , Riemannian manifold , valid covariance

Vol.24 • 2019
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