Electronic Communications in Probability

The Intrinsic geometry of some random manifolds

Sunder Ram Krishnan, Jonathan E. Taylor, and Robert J. Adler

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We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.

Article information

Electron. Commun. Probab. Volume 22 (2017), paper no. 1, 12 pp.

Received: 16 December 2015
Accepted: 25 November 2016
First available in Project Euclid: 5 January 2017

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Digital Object Identifier

Primary: 60G15: Gaussian processes 57N35: Embeddings and immersions 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G60: Random fields 70G45: Differential-geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) [See also 53Cxx, 53Dxx, 58Axx]

Gaussian process manifold random embedding intrinsic functional asymptotics

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Krishnan, Sunder Ram; Taylor, Jonathan E.; Adler, Robert J. The Intrinsic geometry of some random manifolds. Electron. Commun. Probab. 22 (2017), paper no. 1, 12 pp. doi:10.1214/16-ECP4763. https://projecteuclid.org/euclid.ecp/1483585770.

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