Maxima of entries of Haar distributed matrices

Tiefeng Jiang

doi:10.1007/s00440-004-0376-5

Maxima of entries of Haar distributed matrices

Tiefeng Jiang

Statistics (Twin Cities)

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Abstract

Let Γ_n = (γ_ij) be an x n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also W_n := max_{1≤i, j≤n} |γ_ij|. We obtain the limiting distribution and a strong limit theorem on W_n. A tool has been developed to prove these results. It says that up to n/(log n)² columns of Γ_n can be approximated simultaneously by those of some Y_n = (y_ij) in which y_ij are independent standard normals. Similar results are derived also for the unitary group U(n), the special orthogonal group SO(n), and the special unitary group SU(n). -

Original language	English (US)
Pages (from-to)	121-144
Number of pages	24
Journal	Probability Theory and Related Fields
Volume	131
Issue number	1
DOIs	https://doi.org/10.1007/s00440-004-0376-5
State	Published - Jan 2005

Keywords

Gram-Schmidt procedure
Haar measure
Large deviations
Maxima

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title = "Maxima of entries of Haar distributed matrices",

abstract = "Let Γn = (γij) be an x n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also Wn := max1≤i, j≤n |γij|. We obtain the limiting distribution and a strong limit theorem on Wn. A tool has been developed to prove these results. It says that up to n/(log n)2 columns of Γn can be approximated simultaneously by those of some Yn = (yij) in which yij are independent standard normals. Similar results are derived also for the unitary group U(n), the special orthogonal group SO(n), and the special unitary group SU(n). -",

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N2 - Let Γn = (γij) be an x n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also Wn := max1≤i, j≤n |γij|. We obtain the limiting distribution and a strong limit theorem on Wn. A tool has been developed to prove these results. It says that up to n/(log n)2 columns of Γn can be approximated simultaneously by those of some Yn = (yij) in which yij are independent standard normals. Similar results are derived also for the unitary group U(n), the special orthogonal group SO(n), and the special unitary group SU(n). -

AB - Let Γn = (γij) be an x n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also Wn := max1≤i, j≤n |γij|. We obtain the limiting distribution and a strong limit theorem on Wn. A tool has been developed to prove these results. It says that up to n/(log n)2 columns of Γn can be approximated simultaneously by those of some Yn = (yij) in which yij are independent standard normals. Similar results are derived also for the unitary group U(n), the special orthogonal group SO(n), and the special unitary group SU(n). -

KW - Gram-Schmidt procedure

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KW - Large deviations

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Maxima of entries of Haar distributed matrices

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