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). - © Springer-Verlag 2004.