Central limit theorems for the brownian motion on large unitary groups
Bulletin de la Société Mathématique de France, Volume 139 (2011) no. 4, pp. 593-610.

In this paper, we are concerned with the large n limit of the distributions of linear combinations of the entries of a Brownian motion on the group of n×n unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of the asymptotic Gaussian feature of the entries of Haar distributed random unitary matrices, a result already proved by Diaconis et al.

Dans cet article, on considère la loi limite, lorsque n tend vers l’infini, de combinaisons linéaires des coefficients d’un mouvement Brownien sur le groupe des matrices unitaires n×n. On prouve que le processus d’une telle combinaison linéaire converge vers un processus gaussien. Différentes échelles de temps et différentes lois initiales sont considérées, donnant lieu à plusieurs processus limites, liés à la construction géométrique du mouvement Brownien unitaire. En application, on propose une preuve très courte du caractère asymptotiquement gaussien des coefficients d’une matrice unitaire distribuée selon la mesure de Haar, un résultat déjà prouvé par Diaconis et al.

DOI: 10.24033/bsmf.2621
Classification: 15A52, 60B15, 60F05, 46L54
Keywords: unitary brownian motion, heat kernel, random matrices, central limit theorem, Haar measure
Mot clés : mouvement brownien unitaire, noyau de la chaleur, matrices aléatoires, théorème central limite, mesure de Haar
@article{BSMF_2011__139_4_593_0,
     author = {Benaych-Georges, Florent},
     title = {Central limit theorems for the brownian motion on large unitary groups},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {593--610},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {139},
     number = {4},
     year = {2011},
     doi = {10.24033/bsmf.2621},
     mrnumber = {2869307},
     zbl = {1242.60007},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/bsmf.2621/}
}
TY  - JOUR
AU  - Benaych-Georges, Florent
TI  - Central limit theorems for the brownian motion on large unitary groups
JO  - Bulletin de la Société Mathématique de France
PY  - 2011
SP  - 593
EP  - 610
VL  - 139
IS  - 4
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/bsmf.2621/
DO  - 10.24033/bsmf.2621
LA  - en
ID  - BSMF_2011__139_4_593_0
ER  - 
%0 Journal Article
%A Benaych-Georges, Florent
%T Central limit theorems for the brownian motion on large unitary groups
%J Bulletin de la Société Mathématique de France
%D 2011
%P 593-610
%V 139
%N 4
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/bsmf.2621/
%R 10.24033/bsmf.2621
%G en
%F BSMF_2011__139_4_593_0
Benaych-Georges, Florent. Central limit theorems for the brownian motion on large unitary groups. Bulletin de la Société Mathématique de France, Volume 139 (2011) no. 4, pp. 593-610. doi : 10.24033/bsmf.2621. http://archive.numdam.org/articles/10.24033/bsmf.2621/

[1] G. W. Anderson, A. Guionnet & O. Zeitouni - An introduction to random matrices, Cambridge Studies in Advanced Math., vol. 118, Cambridge Univ. Press, 2010. | MR | Zbl

[2] F. Benaych-Georges & T. Lévy - « A continuous semigroup of notions of independence between the classical and the free one », Ann. Probab. 39 (2011), p. 904-938. | MR | Zbl

[3] P. Biane - « Free Brownian motion, free stochastic calculus and random matrices », in Free probability theory (Waterloo, ON, 1995), Fields Inst. Commun., vol. 12, Amer. Math. Soc., 1997, p. 1-19. | MR | Zbl

[4] -, « Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems », J. Funct. Anal. 144 (1997), p. 232-286. | MR | Zbl

[5] É. Borel - « Sur les principes de la théorie cinétique des gaz », Ann. Sci. École Norm. Sup. 23 (1906), p. 9-32. | JFM | Numdam

[6] S. Chatterjee & E. Meckes - « Multivariate normal approximation using exchangeable pairs », ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), p. 257-283. | MR | Zbl

[7] L. H. Y. Chen - « Two central limit problems for dependent random variables », Z. Wahrsch. Verw. Gebiete 43 (1978), p. 223-243. | MR | Zbl

[8] B. Collins, J. A. Mingo, P. Śniady & R. Speicher - « Second order freeness and fluctuations of random matrices. III. Higher order freeness and free cumulants », Doc. Math. 12 (2007), p. 1-70. | MR | Zbl

[9] B. Collins & M. Stolz - « Borel theorems for random matrices from the classical compact symmetric spaces », Ann. Probab. 36 (2008), p. 876-895. | MR | Zbl

[10] A. D'Aristotile, P. Diaconis & C. M. Newman - « Brownian motion and the classical groups », in Probability, statistics and their applications: papers in honor of Rabi Bhattacharya, IMS Lecture Notes Monogr. Ser., vol. 41, Inst. Math. Statist., 2003, p. 97-116. | MR | Zbl

[11] A. Dembo & O. Zeitouni - Large deviations techniques and applications, second éd., Applications of Mathematics (New York), vol. 38, Springer, 1998. | MR | Zbl

[12] N. Demni - « Free Jacobi process », J. Theoret. Probab. 21 (2008), p. 118-143. | MR | Zbl

[13] P. Diaconis & M. Shahshahani - « On the eigenvalues of random matrices », J. Appl. Probab. 31A (1994), p. 49-62. | MR | Zbl

[14] P. Friz & H. Oberhauser - « Rough path limits of the Wong-Zakai type with a modified drift term », J. Funct. Anal. 256 (2009), p. 3236-3256. | MR | Zbl

[15] G. A. Hunt - « Semi-groups of measures on Lie groups », Trans. Amer. Math. Soc. 81 (1956), p. 264-293. | MR | Zbl

[16] N. Ikeda & S. Watanabe - Stochastic differential equations and diffusion processes, second éd., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., 1989. | MR | Zbl

[17] T. Jiang - « How many entries of a typical orthogonal matrix can be approximated by independent normals? », Ann. Probab. 34 (2006), p. 1497-1529. | MR | Zbl

[18] T. Lévy - « Schur-Weyl duality and the heat kernel measure on the unitary group », Adv. Math. 218 (2008), p. 537-575. | MR | Zbl

[19] T. Lévy & M. Maïda - « Central limit theorem for the heat kernel measure on the unitary group », J. Funct. Anal. 259 (2010), p. 3163-3204. | MR | Zbl

[20] E. Meckes - « Linear functions on the classical matrix groups », Trans. Amer. Math. Soc. 360 (2008), p. 5355-5366. | MR | Zbl

[21] J. A. Mingo & A. Nica - « Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices », Int. Math. Res. Not. 2004 (2004), p. 1413-1460. | MR | Zbl

[22] J. A. Mingo, P. Śniady & R. Speicher - « Second order freeness and fluctuations of random matrices. II. Unitary random matrices », Adv. Math. 209 (2007), p. 212-240. | MR | Zbl

[23] J. A. Mingo & R. Speicher - « Second order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces », J. Funct. Anal. 235 (2006), p. 226-270. | MR | Zbl

[24] G. C. Papanicolaou, D. W. Stroock & S. R. S. Varadhan - « Martingale approach to some limit theorems », in Papers from the Duke Turbulence Conference (Duke Univ., Durham, N.C., 1976), Paper No. 6, Duke Univ. Math. Ser., vol. III, Duke Univ., 1977. | MR | Zbl

[25] E. M. Rains - « Combinatorial properties of Brownian motion on the compact classical groups », J. Theoret. Probab. 10 (1997), p. 659-679. | MR | Zbl

[26] L. C. G. Rogers & D. Williams - Diffusions, Markov processes, and martingales. Vol. 2, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., 1987. | MR | Zbl

[27] W. Schneller - « A short proof of Motoo's combinatorial central limit theorem using Stein's method », Probab. Theory Related Fields 78 (1988), p. 249-252. | MR | Zbl

[28] D. W. Stroock & S. R. S. Varadhan - « Limit theorems for random walks on Lie groups », Sankhyā Ser. A 35 (1973), p. 277-294. | MR | Zbl

[29] F. Xu - « A random matrix model from two-dimensional Yang-Mills theory », Comm. Math. Phys. 190 (1997), p. 287-307. | MR | Zbl

Cited by Sources: