Annals of Probability

How many entries of a typical orthogonal matrix can be approximated by independent normals?

Tiefeng Jiang

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We solve an open problem of Diaconis that asks what are the largest orders of pn and qn such that Zn, the pn×qn upper left block of a random matrix Γn which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods.

First, we show that the variation distance between the joint distribution of entries of Zn and that of pnqn independent standard normals goes to zero provided $p_{n}=o(\sqrt{n}\,)$ and $q_{n}=o(\sqrt{n}\,)$. We also show that the above variation distance does not go to zero if $p_{n}=[x\sqrt{n}\,]$ and $q_{n}=[y\sqrt{n}\,]$ for any positive numbers x and y. This says that the largest orders of pn and qn are o(n1/2) in the sense of the above approximation.

Second, suppose Γn=(γij)n×n is generated by performing the Gram–Schmidt algorithm on the columns of Yn=(yij)n×n, where {yij;1≤i,jn} are i.i.d. standard normals. We show that $\varepsilon _{n}(m):=\max_{1\leq i\leq n,1\leq j\leq m}|\sqrt{n}\cdot\gamma_{ij}-y_{ij}|$ goes to zero in probability as long as m=mn=o(n/logn). We also prove that $\varepsilon _{n}(m_{n})\to 2\sqrt{\alpha}$ in probability when mn=[nα/logn] for any α>0. This says that mn=o(n/logn) is the largest order such that the entries of the first mn columns of Γn can be approximated simultaneously by independent standard normals.

Article information

Ann. Probab., Volume 34, Number 4 (2006), 1497-1529.

First available in Project Euclid: 19 September 2006

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Primary: 15A52 60B10: Convergence of probability measures 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60F05: Central limit and other weak theorems 60F99: None of the above, but in this section 62H10: Distribution of statistics

Haar measure Gram–Schmidt algorithm large deviation maxima product distribution random matrix theory variation distance


Jiang, Tiefeng. How many entries of a typical orthogonal matrix can be approximated by independent normals?. Ann. Probab. 34 (2006), no. 4, 1497--1529. doi:10.1214/009117906000000205.

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  • Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis, 2nd ed. Wiley, New York.
  • Apostol, T. M. (1974). Mathematical Analysis, 2nd ed. Addison–Wesley, Reading, MA.
  • Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611–677.
  • Billingsley, P. (1979). Probability and Measure. Wiley, New York.
  • Borel, E. (1906). Introduction géometrique á quelques théories physiques. Gauthier–Villars, Paris. JFM 45.0808.10
  • Chow, Y. S. and Teicher, H. (1988). Probability Theory, Independence, Interchangeability, Martingales, 2nd ed. Springer, New York.
  • Collins, B. (2003). Intégrales matricielles et probabilitiés non-commutatives. Thèse de Doctorat, Univ. Paris 6.
  • D'Aristotle, A., Diaconis, P. and Newman, C. M. (2002). Brownian motion and the classical groups. Probability, Statistics and Their Applications: Papers in Honor of Rabi Bhattacharya 97–116. IMS Lecture Notes Monogr. Ser. 41. IMS, Beachwood, OH.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Diaconis, P. (2003). Patterns in eigenvalues: The 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. (N.S.) 40 155–178 (electronic).
  • Diaconis, P. and Freedman, D. (1987). A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist. 23 397–423.
  • Diaconis, P. and Shahshahni, M. (1994). On the eigenvalues of random matrices. Studies in applied probability. J. Appl. Probab. 31A 49–62.
  • Diaconis, P. and Evans, S. N. (2001). Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 2615–2633 (electronic).
  • Diaconis, P. W., Eaton, M. L. and Lauritzen, S. L. (1992). Finite de Finetti theorems in linear models and multivariate analysis. Scand. J. Statist. 19 289–315.
  • Eaton, M. L. (1989). Group Invariance Applications in Statistics. IMS, Hayward, CA.
  • Gallardo, L. (1983). Au sujet du contenu probabiliste d'un lemma d'Henri Poincaré. Ann. Univ. Clemont 69 192–197.
  • Geman, S. (1980). A limit theorem for the norm of random matrices. Ann. Probab. 8 252–261.
  • Horn, R. and Johnson, C. (1990). Matrix Analysis. Cambridge Univ. Press.
  • Jiang, T. (2005). Maxima of entries of Haar distributed matrices. Probab. Theory Related Fields 131 121–144.
  • Johansson, K. (1997). On random matrices from the compact classical groups. Ann. of Math. (2) 145 519–545.
  • Jonsson, D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1–38.
  • Lang, S. (1987). Calculus of Several Variables. Springer, New York.
  • Letac, G. (1981). Isotropy and sphericity: Some characterisations of the normal distribution. Ann. Statist. 9 408–417.
  • Maxwell, J. C. (1875). Theory of Heat, 4th ed. Longmans, London.
  • Maxwell, J. C. (1878). On Boltzmann's theorem on the average distribution of energy in a system of material points. Cambridge Phil. Soc. Trans. 12 547. JFM 11.0776.01
  • McKean, H. P. (1973). Geometry of differential space. Ann. Probab. 1 197–206.
  • Poincaré, H. (1912). Calcul des probabilitiés. Gauthier–Villars, Paris. JFM 43.0308.04
  • Rains, E. M. (1997). High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107 219–241.
  • Rao, C. R. (1973). Linear Statistical Inference and Its Applications. Wiley, New York.
  • Stam, A. J. (1982). Limit theorems for uniform distributions on spheres in high-dimensional Euclidean spaces. J. Appl. Probab. 19 221–228.
  • Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. (1988). On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theory Related Fields 78 509–521.
  • Yor, M. (1985). Inégalitiés de martingales continus arrêtès à un temps quelconques I. Lecture Notes in Math. 1118. Springer, Berlin. Zbl 0563.60045