Stein's Method and the Multivariate CLT for Traces of Powers on the Compact Classical Groups

Christian Döbler (Ruhr University Bochum)
Michael Stolz (Ruhr University Bochum)

Abstract


Let $M$ be a random element of the unitary, special orthogonal, or unitary symplectic groups, distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large matrix size $n$, the vector of traces of consecutive powers of $M$ tends to a vector of independent (real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single power $j$ (which may grow with $n$), a speed of convergence result may be obtained via Stein's method of exchangeable pairs. In this note, we extend Fulman's result to the multivariate central limit theorem for the full vector of traces of powers.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 2375-2405

Publication Date: November 22, 2011

DOI: 10.1214/EJP.v16-960

References

  1. S. Chatterjee, E. Meckes. Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 257--283. MR2453473 (2010c:60072)
  2. L.H.Y. Chen, Q.M. Shao. Stein's method for normal approximation. An introduction to Stein's method, 1--59, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005. MR2235448
  3. P. Diaconis, M. Shahshahani. On the eigenvalues of random matrices. Studies in applied probability. J. Appl. Probab. 31A (1994), 49--62. MR1274717 (95m:60011)
  4. J. Fulman. Stein's method, heat kernel, and traces of powers of elements of compact Lie groups, 2010. Available on arXiv.org:1005.1306.
  5. S. Helgason. Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. Corrected reprint of the 1984 original. Mathematical Surveys and Monographs, 83. American Mathematical Society, Providence, RI, 2000. xxii+667 pp. ISBN: 0-8218-2673-5 MR1790156 (2001h:22001)
  6. C.P. Hughes, Z. Rudnick. Mock-Gaussian behaviour for linear statistics of classical compact groups. Random matrix theory. J. Phys. A 36 (2003), no. 12, 2919--2932. MR1986399 (2004e:60012)
  7. N. Ikeda, S. Watanabe. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3 MR1011252 (90m:60069)
  8. K. Johansson. On random matrices from the compact classical groups. Ann. of Math. (2) 145 (1997), no. 3, 519--545. MR1454702 (98e:60016)
  9. T. LÈvy. Schur-Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218 (2008), no. 2, 537--575. MR2407946 (2009g:15075)
  10. E. Meckes. On Stein's method for multivariate normal approximation. High dimensional probability V: the Luminy volume, 153--178, Inst. Math. Stat. Collect., 5, Inst. Math. Statist., Beachwood, OH, 2009. MR2797946
  11. L. Pastur, V. Vasilchuk. On the moments of traces of matrices of classical groups. Comm. Math. Phys. 252 (2004), no. 1-3, 149--166. MR2104877 (2005j:60010)
  12. E.M. Rains. Combinatorial properties of Brownian motion on the compact classical groups. J. Theoret. Probab. 10 (1997), no. 3, 659--679. MR1468398 (99f:60016)
  13. Y. Rinott, V. Rotar. On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics. Ann. Appl. Probab. 7 (1997), no. 4, 1080--1105. MR1484798 (99g:60050)
  14. G. Reinert, A. R?llin. Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition. Ann. Probab. 37 (2009), no. 6, 2150--2173. MR2573554 (2011e:60047)
  15. E.M. Stein. Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No. 63 Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1970 viii+146 pp. MR0252961 (40 #6176)
  16. C. Stein. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, pp. 583--602. Univ. California Press, Berkeley, Calif., 1972. MR0402873 (53 #6687)
  17. C. Stein. Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. iv+164 pp. ISBN: 0-940600-08-0 MR0882007 (88j:60055)
  18. C. Stein, The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Department of Statistics, Stanford University, Technical Report No. 470, 1995
  19. M. Stolz. On the Diaconis-Shahshahani method in random matrix theory. J. Algebraic Combin. 22 (2005), no. 4, 471--491. MR2191648 (2007i:15036)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.