Euclidean Distance Between Haar Orthogonal and Gaussian Matrices

@article{GonzlezGuilln2014EuclideanDB,
  title={Euclidean Distance Between Haar Orthogonal and Gaussian Matrices},
  author={Carlos E. Gonz{\'a}lez-Guill{\'e}n and Carlos Palazuelos and Ignacio Villanueva},
  journal={Journal of Theoretical Probability},
  year={2014},
  volume={31},
  pages={93-118}
}
In this work, we study a version of the general question of how well a Haar-distributed orthogonal matrix can be approximated by a random Gaussian matrix. Here, we consider a Gaussian random matrix $$Y_n$$Yn of order n and apply to it the Gram–Schmidt orthonormalization procedure by columns to obtain a Haar-distributed orthogonal matrix $$U_n$$Un. If $$F_i^m$$Fim denotes the vector formed by the first m-coordinates of the ith row of $$Y_n-\sqrt{n}U_n$$Yn-nUn and $$\alpha \,=\,\frac{m}{n}$$α=mn… 

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