Abstract.
Let Γ n =(γ ij ) be an n×n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also W n :=max1≤ 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).
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Andersen, H., Højbjerre, M., Sørensen, D., Erikson, P.: Linear and Graphical Models for the Multivariate Complex Normal Distributions. Springer-Verlag, 1995
Anderson, T.W.: An Introduction to Multivariate Statistical Analysis. Second edition, John Wiley and Sons, 1984
Arratia, R., Goldstein, L., Gordon, L.: Two Moments Suffice for Poisson Approximation: the Chen-Stein Method. Ann. Probab. 17, 9–25 (1989)
Borel, E.: Sur les Principes de la theorie cinetique des gaz. Annales, L’Ecole Normal Sup. 23, 9–32 (1906)
Borodin, A., Olshansky, G.: Correlation kernels arising from the infinite-dimension unitary group and its representations. Technical report, University of Pennsylvania, Department of Mathematics, 2001
Chow, Y.S., Teicher, H.: Probability Theory, Independence, Interchangeability, Martingales. Third edition, Springer Texts in Statistics, 1997
D’Aristotile, A., Diaconis, P., Newman, C.: Brownian motion and the classical groups. Technical report, Department of Statistics, Stanford University, 2002
D’Aristotile, A., Diaconis, P., Freedman, D.: On merging of probabilities. Sankhya, Series A 50, 363–380 (1988)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Second edition, Springer, 1998
Diaconis, P.: Patterns in eigenvalues: the 70th Willard Gibbs lectures. Bulletin of the American Mathematical Society 40, 155–178 (2003)
Diaconis, P., Eaton, M., Lauritzen, L.: Finite deFinetti theorems in linear models and multivariate analysis. Scand. J. Statist. 19 (4), 289–315 (1992)
Diaconis, P., Evans, S.: Linear functionals of eigenvalues of random matrices. Trans. Am. Math. Soc. 353, 2615–2633 (2001)
Diaconis, P., Freedman, D.: A dozen deFinetti-style results in search of a theory. Ann.Inst. Henri Poincaré 23, 397–423 (1987)
Donoho, D., Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory 47 (7), 289–315 (2001)
Durrett, R.: Probability: Theory and Examples. Second edition, The Duxbury Press, 1995
Eaton, M.: Multivariate Statistics, A Vector Space Approach. Wiley, New York, 1983
Eaton, M.: Group-Invariance Applications in Statistics. Regional Conference Series in Probability and Statistics, Vol. 1. IMS, Hayward, California, 1989
Gnedenko, B.: Sur la distribution limite du terme maximum d’une serie aleatoire. Ann. Math. 44 (3), 423–453 (1943)
Grove, L.: Classical Groups and Geometric Algebra (Graduate Studies in Mathematics, 39), American Mathematical Society, 2001
Horn, R. and Johnson, C.: Matrix Analysis. Cambridge University Press, 1999
Jiang, T.: Maxima of partial sums indexed by geometrical structures. Ann. Probab. 30 (4), 1854–1892 (2002)
Jiang, T.: A comparison of scores of two protein structures with foldings. Ann. Probab. 30 (4), 1893–1912 (2002)
Jiang, T.: The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14 (2), 865–880 (2004)
Jiang, T.: How many entries of a typical orthogonal matrix can be approximated by independent normals? I. Technical report, School of Statistics, University of Minnesota, 2003
Jiang, T.: How many entries of a typical orthogonal matrix can be approximated by independent normals? II. Technical report, School of Statistics, University of Minnesota, 2003
Johansson, K.: On random matrices from the compact classical groups. Ann. Math. 145, 519–545 (1997)
Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, 1983
Ledoux, M. and Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer-Verlag, New York, 1991
Mheta, M.L.: Random Matrices. 2nd edition, Academic Press, Boston, 1991
Nachbin, L.: The Haar Integral. Krieger Publ. Co., Huntington, New York, 1976
Olshanski, G., Vershik, A.: Ergodic unitarily invariant measures on the space of infinite Hermitian matrices. Contemporary Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, 1996, pp. 137–175
Pickrell, D.: Mackey analysis of infinite classical motion groups. Pacific J. 150, 139–166 (1991)
Resnick, S.: Extreme Values, Regular Variations, and Point Processes. Springer-Verlag, New York, 1987
Rao, C.R.: Linear Statistical Inference and Its Applications. Second edition, John Wiley and Sons, Inc., 1973
Weyl, H.: The Classical Groups. Princeton University Press, 1946
Wijsman, R.: Invariant Measures on Groups and Their Use in Statistics. IMA, Lecture Notes-Monograph Series, Hayward, California, 1990
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000):15A52, 60B10, 60B15, 60F10
Rights and permissions
About this article
Cite this article
Jiang, T. Maxima of entries of Haar distributed matrices. Probab. Theory Relat. Fields 131, 121–144 (2005). https://doi.org/10.1007/s00440-004-0376-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0376-5