Almost sure and $L^k$-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
References
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. 0949.60077Bai, 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. 0949.60077
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. 0652.60040 10.1016/0047-259X(88)90078-4Bai, 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. 0652.60040 10.1016/0047-259X(88)90078-4
Bru, Marie-France. Diffusions of perturbed principal component analysis. J. Multivariate Anal. 29 (1989), no. 1, 127–136. 0687.62048 10.1016/0047-259X(89)90080-8Bru, Marie-France. Diffusions of perturbed principal component analysis. J. Multivariate Anal. 29 (1989), no. 1, 127–136. 0687.62048 10.1016/0047-259X(89)90080-8
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 1022.60026 10.1214/aop/1015345602 euclid.aop/1015345602Cabanal 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 1022.60026 10.1214/aop/1015345602 euclid.aop/1015345602
Capitaine, M.; Donati-Martin, C. Free Wishart processes. J. Theoret. Probab. 18 (2005), no. 2, 413–438. 1084.46051 10.1007/s10959-005-3511-zCapitaine, M.; Donati-Martin, C. Free Wishart processes. J. Theoret. Probab. 18 (2005), no. 2, 413–438. 1084.46051 10.1007/s10959-005-3511-z
Demni, Nizar. The Laguerre process and generalized Hartman-Watson law. Bernoulli 13 (2007), no. 2, 556–580. 1139.60037 10.3150/07-BEJ6048 euclid.bj/1179498761Demni, Nizar. The Laguerre process and generalized Hartman-Watson law. Bernoulli 13 (2007), no. 2, 556–580. 1139.60037 10.3150/07-BEJ6048 euclid.bj/1179498761
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. 1076.60067 10.2977/prims/1145475450Donati-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. 1076.60067 10.2977/prims/1145475450
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 MR838085Ethier, 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 MR838085
Geman, Stuart. A limit theorem for the norm of random matrices. Ann. Probab. 8 (1980), no. 2, 252–261. 0428.60039 10.1214/aop/1176994775 euclid.aop/1176994775Geman, Stuart. A limit theorem for the norm of random matrices. Ann. Probab. 8 (1980), no. 2, 252–261. 0428.60039 10.1214/aop/1176994775 euclid.aop/1176994775
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. 1067.60003 10.1007/s10959-004-0579-9Graczyk, 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. 1067.60003 10.1007/s10959-004-0579-9
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 1131.60008 10.1137/S0040585X97982682Graczyk, 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 1131.60008 10.1137/S0040585X97982682
A Guionnet.: Random Matrices: Lectures on MacroscopicAsymptotics. Ecole d'Èté des Probabilites de Saint-Flour XXXVI2006, (Lecture Notes in Mathematics), Springer, 2008.A Guionnet.: Random Matrices: Lectures on MacroscopicAsymptotics. Ecole d'Èté des Probabilites de Saint-Flour XXXVI2006, (Lecture Notes in Mathematics), Springer, 2008.
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: 0-8218-2081-8Hiai, 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
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: 3-540-17882-1Jacod, 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
Katori, Makoto; Tanemura, Hideki. Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45 (2004), no. 8, 3058–3085. 1071.82045 10.1063/1.1765215Katori, Makoto; Tanemura, Hideki. Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45 (2004), no. 8, 3058–3085. 1071.82045 10.1063/1.1765215
König, Wolfgang; O'Connell, Neil. Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Electron. Comm. Probab. 6 (2001), 107–114 (electronic).König, Wolfgang; O'Connell, Neil. Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Electron. Comm. Probab. 6 (2001), 107–114 (electronic).
Lawi, Stephan. Hermite and Laguerre polynomials and matrix-valued stochastic processes. Electron. Commun. Probab. 13 (2008), 67–84. 1189.60144 10.1214/ECP.v13-1353Lawi, Stephan. Hermite and Laguerre polynomials and matrix-valued stochastic processes. Electron. Commun. Probab. 13 (2008), 67–84. 1189.60144 10.1214/ECP.v13-1353
Marchenko, V. A.; Pastur, L. A. Distribution of eigenvalues in certain sets of random matrices.(Russian) Mat. Sb. (N.S.) 72 (114) 1967 507–536. MR0208649Marchenko, 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
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.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.
Silverstein, Jack W. The smallest eigenvalue of a large-dimensional Wishart matrix. Ann. Probab. 13 (1985), no. 4, 1364–1368. 0591.60025 10.1214/aop/1176992819 euclid.aop/1176992819Silverstein, Jack W. The smallest eigenvalue of a large-dimensional Wishart matrix. Ann. Probab. 13 (1985), no. 4, 1364–1368. 0591.60025 10.1214/aop/1176992819 euclid.aop/1176992819
Wachter, Kenneth W. The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probability 6 (1978), no. 1, 1–18. 0374.60039 10.1214/aop/1176995607 euclid.aop/1176995607Wachter, Kenneth W. The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probability 6 (1978), no. 1, 1–18. 0374.60039 10.1214/aop/1176995607 euclid.aop/1176995607