May 2020 Random orthogonal matrices and the Cayley transform
Michael Jauch, Peter D. Hoff, David B. Dunson
Bernoulli 26(2): 1560-1586 (May 2020). DOI: 10.3150/19-BEJ1176

## Abstract

Random orthogonal matrices play an important role in probability and statistics, arising in multivariate analysis, directional statistics, and models of physical systems, among other areas. Calculations involving random orthogonal matrices are complicated by their constrained support. Accordingly, we parametrize the Stiefel and Grassmann manifolds, represented as subsets of orthogonal matrices, in terms of Euclidean parameters using the Cayley transform. We derive the necessary Jacobian terms for change of variables formulas. Given a density defined on the Stiefel or Grassmann manifold, these allow us to specify the corresponding density for the Euclidean parameters, and vice versa. As an application, we present a Markov chain Monte Carlo approach to simulating from distributions on the Stiefel and Grassmann manifolds. Finally, we establish that the Euclidean parameters corresponding to a uniform orthogonal matrix can be approximated asymptotically by independent normals. This result contributes to the growing literature on normal approximations to the entries of random orthogonal matrices or transformations thereof.

## Citation

Michael Jauch. Peter D. Hoff. David B. Dunson. "Random orthogonal matrices and the Cayley transform." Bernoulli 26 (2) 1560 - 1586, May 2020. https://doi.org/10.3150/19-BEJ1176

## Information

Received: 1 March 2019; Revised: 1 November 2019; Published: May 2020
First available in Project Euclid: 31 January 2020

zbMATH: 07166574
MathSciNet: MR4058378
Digital Object Identifier: 10.3150/19-BEJ1176

Keywords: Gaussian approximation , Grassmann manifold , Jacobian , Markov chain Monte Carlo , Stiefel manifold

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

JOURNAL ARTICLE
27 PAGES

Vol.26 • No. 2 • May 2020