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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 76 open journal systems 


On The Traces Of Laguerre Processes

Victor Perez Abreu, Centro de Investigacion en Matematicas
Constantin Tudor, University of Bucharest


Abstract
Almost sure and Lk-convergence of the traces of Laguerre processes to the family of dilations of the standard free Poisson distribution are established. We also prove that the fluctuations around the limiting process, converge weakly to a continuous centered Gaussian process. The almost sure convergence on compact time intervals of the largest and smallest eigenvalues processes is also established


Full text: PDF

Pages: 2241-2263

Published on: October 6, 2009


Bibliography
  1. Bai, Z. D. Methodologies in spectral analysis of large-dimensional random matrices, a review.With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. Statist. Sinica 9 (1999), no. 3, 611--677. MR1711663 (2000e:60044)
  2. Bai, Z. D.; Yin, Y. Q. Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix. Ann. Probab. 21 (1993), no. 3, 1275--1294. MR1235416 (94j:60060)
  3. Bai, Z. D.; Silverstein, Jack W.; Yin, Y. Q. A note on the largest eigenvalue of a large-dimensional sample covariance matrix. J. Multivariate Anal. 26 (1988), no. 2, 166--168. MR0963829 (89i:62083)
  4. Bru, Marie-France. Diffusions of perturbed principal component analysis. J. Multivariate Anal. 29 (1989), no. 1, 127--136. MR0991060 (90k:62123)
  5. Cabanal Duvillard, T.; Guionnet, A. Large deviations upper bounds for the laws of matrix-valued processes and non-communicative entropies. Ann. Probab. 29 (2001), no. 3, 1205--1261. MR1872742 (2003a:60040)
  6. Capitaine, M.; Donati-Martin, C. Free Wishart processes. J. Theoret. Probab. 18 (2005), no. 2, 413--438. MR2137451 (2006e:46072)
  7. Demni, Nizar. The Laguerre process and generalized Hartman-Watson law. Bernoulli 13 (2007), no. 2, 556--580. MR2331264 (2008e:60257)
  8. Demni N.: Processus Stochastiques Matriciels, Systemesde Racines et Probabilites Non Commutatives. Thesis, Universite Pierre et Marie Curie, 2007.
  9. Donati-Martin, Catherine; Doumerc, Yan; Matsumoto, Hiroyuki; Yor, Marc. Some properties of the Wishart processes and a matrix extension of the Hartman-Watson laws. Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, 1385--1412. MR2105711 (2005k:60251)
  10. Ethier, Stewart N.; Kurtz, Thomas G. Markov processes.Characterization and convergence.Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085 (88a:60130)
  11. Haagerup, Uffe; Thorbjørnsen, Steen. Random matrices with complex Gaussian entries. Expo. Math. 21 (2003), no. 4, 293--337. MR2022002 (2005b:46142)
  12. Geman, Stuart. A limit theorem for the norm of random matrices. Ann. Probab. 8 (1980), no. 2, 252--261. MR0566592 (81m:60046)
  13. Graczyk, Piotr; Letac, Gérard; Massam, Hélène. The complex Wishart distribution and the symmetric group. Ann. Statist. 31 (2003), no. 1, 287--309. MR1962508 (2003m:62166)
  14. Graczyk, P.; Letac, G.; Massam, H. The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution. J. Theoret. Probab. 18 (2005), no. 1, 1--42. MR2132270 (2006e:60021)
  15. Graczyk, P.; Vostrikova, L. The moments of Wishart processes via Itô calculus. Teor. Veroyatn. Primen. 51 (2006), no. 4, 732--751; translation in Theory Probab. Appl. 51 (2007), no. 4, 609--625 MR2338064 (2008k:62026)
  16. A Guionnet.: Random Matrices: Lectures on MacroscopicAsymptotics.Ecole d'été des Probabilites de Saint-Flour XXXVI2006, (Lecture Notes in Mathematics), Springer, 2008.
  17. Hiai, Fumio; Petz, Dénes. The semicircle law, free random variables and entropy.Mathematical Surveys and Monographs, 77. American Mathematical Society, Providence, RI, 2000. x+376 pp. ISBN: 0-8218-2081-8 MR1746976 (2001j:46099)
  18. Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1 MR0959133 (89k:60044)
  19. Katori, Makoto; Tanemura, Hideki. Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45 (2004), no. 8, 3058--3085. MR2077500 (2005g:82099)
  20. König, Wolfgang; O'Connell, Neil. Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Electron. Comm. Probab. 6 (2001), 107--114 (electronic). MR1871699 (2002j:15025)
  21. Lawi, Stephan. Hermite and Laguerre polynomials and matrix-valued stochastic processes. Electron. Commun. Probab. 13 (2008), 67--84. MR2386064 (2009a:60090)
  22. Letac, Gérard; Massam, Hélène. All invariant moments of the Wishart distribution. Scand. J. Statist. 31 (2004), no. 2, 295--318. MR2066255 (2005f:62092)
  23. Marčenko, V. A.; Pastur, L. A. Distribution of eigenvalues in certain sets of random matrices.(Russian) Mat. Sb. (N.S.) 72 (114) 1967 507--536. MR0208649 (34 #8458)
  24. Oravecz, Ferenc; Petz, Dénes. On the eigenvalue distribution of some symmetric random matrices. Acta Sci. Math. (Szeged) 63 (1997), no. 3-4, 383--395. MR1480488 (99a:60020)
  25. Pérez-Abreu, Víctor; Tudor, Constantin. Functional limit theorems for trace processes in a Dyson Brownian motion. Commun. Stoch. Anal. 1 (2007), no. 3, 415--428. MR2403859 (2009k:60216)
  26. Silverstein, Jack W. The smallest eigenvalue of a large-dimensional Wishart matrix. Ann. Probab. 13 (1985), no. 4, 1364--1368. MR0806232 (87b:60050)
  27. Voiculescu D., Dykema K.J. and Nica A.: Free RandomVariables. CRM Monographs Series, Vol. 1, 1992. MR1217253
  28. Wachter, Kenneth W. The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probability 6 (1978), no. 1, 1--18. MR0467894 (57 #7744)
















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Electronic Journal of Probability. ISSN: 1083-6489