## Electronic Communications in Probability

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

### The Intrinsic geometry of some random manifolds

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

#### Abstract

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

**Source**

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

**Dates**

Received: 16 December 2015

Accepted: 25 November 2016

First available in Project Euclid: 5 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1483585770

**Digital Object Identifier**

doi:10.1214/16-ECP4763

**Mathematical Reviews number (MathSciNet)**

MR3607796

**Zentralblatt MATH identifier**

1366.57012

**Subjects**

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]

**Keywords**

Gaussian process manifold random embedding intrinsic functional asymptotics

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

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