The Entries of Haar-Invariant Matrices from the Classical Compact Groups

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Let Γn=(γij)n×n be a random matrix with the Haar probability measure on the orthogonal group O(n), the unitary group U(n), or the symplectic group Sp(n). Given 1≤m < n, a probability inequality for a distance between (γij)n×m and some mn independent F-valued normal random variables is obtained, where F=ℝ, ℂ, or ℍ (the set of real quaternions). The result is universal for the three cases. In particular, the inequality for Sp(n) is new.

Original languageEnglish (US)
Pages (from-to)1227-1243
Number of pages17
JournalJournal of Theoretical Probability
Issue number4
StatePublished - Dec 2010

Bibliographical note

Funding Information:
Supported in part by NSF#DMS-0449365.


  • Classical compact group
  • Gaussian distribution
  • Haar measure
  • Independence
  • Probability inequality
  • Random matrix


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