Brownian sheets; Dyson Brownian motion; Eigenvalue distribution; Fractional Brownian motion; Matrix-valued process; Squared Bessel particle system; Wishart process; Statistics and Probability; Numerical Analysis; Statistics, Probability and Uncertainty
Résumé :
[en] Since the introduction of Dyson's Brownian motion in early 1960s, there have been a lot of developments in the investigation of stochastic processes on the space of Hermitian matrices. Their properties, especially, the properties of their eigenvalues have been studied in great detail. In particular, the limiting behaviours of the eigenvalues are found when the dimension of the matrix space tends to infinity, which connects with random matrix theory. This survey reviews a selection of results on the eigenvalues of stochastic processes from the literature of the past three decades. For most recent variations of such processes, such as matrix-valued processes driven by fractional Brownian motion or Brownian sheet, the eigenvalues of them are also discussed in this survey. In the end, some open problems in the area are also proposed.
Disciplines :
Mathématiques
Auteur, co-auteur :
Song, Jian ; Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, China
Yao, Jianfeng ; Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong
YUAN, Wangjun ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) ; Department of Mathematics and Statistics, University of Ottawa, Canada
Co-auteurs externes :
yes
Langue du document :
Anglais
Titre :
Recent advances on eigenvalues of matrix-valued stochastic processes
The authors would like to thank the Editor, Prof. Dietrich von Rosen, for his invitation to contribute to this special jubilee issue. Comments from Prof. Victor Pérez-Abreu on this review are also acknowledged. Jian Song’s research is partially supported by Shandon University (Grant No. 11140089963041 ) and the National Natural Science Foundation of China (Grant No. 12071256 ). Jianfeng Yao’s research is partially supported by the Research Grant Council of Hong Kong SAR (GRF Grant 17307319 ).
Anderson, G.W., Guionnet, A., Zeitouni, O., An Introduction to Random Matrices. 2010, Cambridge University Press, Cambridge.
Bai, Z., Silverstein, J.W., Spectral Analysis of Large Dimensional Random Matrices, second ed. Springer Series in Statistics, 2010, Springer, New York.
Berman, R.J., Önnheim, M., Propagation of chaos for a class of first order models with singular mean field interactions. SIAM J. Math. Anal. 51 (2019), 159–196.
Biane, P., Free Brownian motion, free stochastic calculus and random matrices. Free Probability Theory (Waterloo, ON, 1995) Fields Institute Communications, vol. 12, 1997, Amer. Math. Soc., Providence, RI, 1–19,.
Biane, P., Speicher, R., Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Related Fields 112 (1998), 373–409.
Bru, M.-F., Diffusions of perturbed principal component analysis. J. Multivariate Anal. 29 (1989), 127–136.
Bru, M.-F., Wishart processes. J. Theoret. Probab. 4 (1991), 725–751.
Cabanal-Duvillard, T., Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. Henri Poincaré Probab. Stat. 37 (2001), 373–402.
Cabanal-Duvillard, T., Guionnet, A., Large deviations upper bounds for the laws of matrix-valued processes and non-communicative entropies. Ann. Probab. 29 (2001), 1205–1261.
Cairoli, R., Walsh, J.B., Stochastic integrals in the plane. Acta Math. 134 (1975), 111–183.
Cépa, E., Lépingle, D., Diffusing particles with electrostatic repulsion. Probab. Theory Related Fields 107 (1997), 429–449.
Chan, T., The Wigner semi-circle law and eigenvalues of matrix-valued diffusions. Probab. Theory Related Fields 93 (1992), 249–272.
Da Fonseca, J., Grasselli, M., Ielpo, F., Estimating the Wishart affine stochastic correlation model using the empirical characteristic function. Stud. Nonlinear Dyn. Econom. 18 (2014), 253–289.
Diaz, M., Jaramillo, A., Pardo, J.C., Fluctuations for matrix-valued Gaussian processes. 2020 arXiv preprint, arXiv:2001.03718.
Dynkin, E.B., Non-negative eigenfunctions of the Laplace-Beltrami operator and Brownian motion in certain symmetric spaces. Dokl. Akad. Nauk SSSR 141 (1961), 288–291.
Dyson, F.J., A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 (1962), 1191–1198.
Fernique, X., Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci., Paris A–B 270 (1970), A1698–A1699.
Fonseca, J.D., Grasselli, M., Tebaldi, C., Option pricing when correlations are stochastic: An analytical framework. Rev. Deriv. Res. 10 (2007), 151–180.
Gnoatto, A., The Wishart short rate model. Int. J. Theor. Appl. Finance, 15, 2012, 1250056 24.
Gnoatto, A., Grasselli, M., An affine multicurrency model with stochastic volatility and stochastic interest rates. SIAM J. Financial Math. 5 (2014), 493–531.
Göing-Jaeschke, A., Yor, M., A survey and some generalizations of Bessel processes. Bernoulli 9 (2003), 313–349.
Gouriéroux, C., Continuous time Wishart process for stochastic risk. Econometric Rev. 25 (2006), 177–217.
Gouriéroux, C., Sufana, R., Derivative pricing with Wishart multivariate stochastic volatility. J. Bus. Econom. Statist. 28 (2010), 438–451.
Graczyk, P., Małecki, J., Multidimensional Yamada-Watanabe theorem and its applications to particle systems. J. Math. Phys., 54, 2013, 021503 15.
Graczyk, P., Małecki, J., Strong solutions of non-colliding particle systems. Electron. J. Probab., 19(119), 2014 no. 119, 21.
Hiai, F., Petz, D., The Semicircle Law, Free Random Variables and Entropy Mathematical Surveys and Monographs, vol. 77, 2000, American Mathematical Society, Providence, RI.
Hu, Y., Nualart, D., Song, J., Fractional martingales and characterization of the fractional Brownian motion. Ann. Probab. 37 (2009), 2404–2430.
Jabin, P.-E., Wang, Z., Quantitative estimates of propagation of chaos for stochastic systems with W−1,∞ kernels. Invent. Math. 214 (2018), 523–591.
Jaramillo, A., Nualart, D., Collision of eigenvalues for matrix-valued processes. Random Matrices Theory Appl., 9, 2020, 2030001 26.
Jaramillo, A., Pardo, J.C., Pérez, J.L., Convergence of the empirical spectral distribution of Gaussian matrix-valued processes. Electron. J. Probab., 24, 2019 Paper No. 10, 22.
Kahn, E., About the eigenvalues of Wishart processes. 2020 arXiv preprint arXiv:2009.09874.
Karatzas, I., Shreve, S.E., Brownian Motion and Stochastic Calculus, second ed. Graduate Texts in Mathematics, vol. 113, 1991, Springer-Verlag, New York.
Katori, M., Bessel Processes, Schramm-Loewner Evolution, and the Dyson Model SpringerBriefs in Mathematical Physics, vol. 11, 2015, Springer, Singapore.
Katori, M., Tanemura, H., Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45 (2004), 3058–3085.
Katori, M., Tanemura, H., Noncolliding Brownian motion and determinantal processes. J. Stat. Phys. 129 (2007), 1233–1277.
Katori, M., Tanemura, H., Noncolliding squared Bessel processes. J. Stat. Phys. 142 (2011), 592–615.
Kendall, W.S., The diffusion of Euclidean shape. Disorder in Physical Systems, 1990, Oxford Sci. Publ., Oxford Univ. Press, New York, 203–217.
Khoshnevisan, D., Multiparameter Processes Springer Monographs in Mathematics, 2002, Springer-Verlag, New York An introduction to random fields.
Kondor, R., Jebara, T., Gaussian and Wishart hyperkernels. Advances in Neural Information Processing Systems, Vol. 19, 2007, MIT Press, 729–736.
König, W., O'Connell, N., Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Electron. Commun. Probab. 6 (2001), 107–114.
Li, S., Li, X.-D., Xie, Y.-X., On the law of large numbers for the empirical measure process of generalized Dyson Brownian motion. J. Stat. Phys. 181 (2020), 1277–1305.
Li, W.-J., Zhang, Z., Yeung, D.-Y., Latent Wishart processes for relational kernel learning. Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics Proceedings of Machine Learning Research, vol. 5, 2009, PMLR, Hilton Clearwater Beach Resort, Clearwater Beach, Florida USA, 336–343.
Li, J., Zhao, B., Deng, C., Xu, R.Y.D., Time varying metric learning for visual tracking. Pattern Recognit. Lett. 80 (2016), 157–164.
Małecki, J., Pérez, J.L., Universality classes for general random matrix flows. 2019 arXiv e-prints, arXiv:1901.02841.
Mayerhofer, E., Pfaffel, O., Stelzer, R., On strong solutions for positive definite jump diffusions. Stochastic Process. Appl. 121 (2011), 2072–2086.
McKean, H.P., Stochastic Integrals. 2005, AMS Chelsea Publishing, Providence, RI Reprint of the 1969 edition, with errata.
Mehta, M.L., Random Matrices, third ed. Pure and Applied Mathematics (Amsterdam), vol. 142, 2004, Elsevier/Academic Press, Amsterdam.
Mingo, J.A., Speicher, R., Free Probability and Random Matrices Fields Institute Monographs, vol. 35, 2017, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON.
Norris, J.R., Rogers, L.C.G., Williams, D., Brownian motions of ellipsoids. Trans. Amer. Math. Soc. 294 (1986), 757–765.
Nourdin, I., Taqqu, M.S., Central and non-central limit theorems in a free probability setting. J. Theoret. Probab. 27 (2014), 220–248.
Nualart, D., The Malliavin Calculus and Related Topics, Probability and Its Applications (New York). second ed., 2006, Springer-Verlag, Berlin.
Nualart, D., Pérez-Abreu, V., On the eigenvalue process of a matrix fractional Brownian motion. Stochastic Process. Appl. 124 (2014), 4266–4282.
Orihara, A., On random ellipsoid. J. Fac. Sci. Univ. Tokyo 17 (1970), 73–85.
Pardo, J.C., Pérez, J.-L., Pérez-Abreu, V., A random matrix approximation for the non-commutative fractional Brownian motion. J. Theoret. Probab. 29 (2016), 1581–1598.
Pardo, J.C., Pérez, J.-L., Pérez-Abreu, V., On the non-commutative fractional Wishart process. J. Funct. Anal. 272 (2017), 339–362.
Pérez-Abreu, V., Tudor, C., Functional limit theorems for trace processes in a Dyson Brownian motion. Commun. Stoch. Anal. 1 (2007), 415–428.
Pérez-Abreu, V., Tudor, C., On the traces of Laguerre processes. Electron. J. Probab., 14, 2009 no. 76, 2241–2263.
Rogers, L.C.G., Shi, Z., Interacting Brownian particles and the Wigner law. Probab. Theory Related Fields 95 (1993), 555–570.
Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives. 1993, Gordon and Breach Science Publishers, Yverdon.
Schölkopf, B., Smola, A.J., Learning with Kernels. 2002, The MIT Press Support Vector Machines, Regularization, Optimization, and Beyond.
Serfaty, S., Mean field limit for Coulomb-type flows. Duke Math. J. 169 (2020), 2887–2935 With an appendix by Mitia Duerinckx and Serfaty.
Song, J., Xiao, Y., Yuan, W., On collision of multiple eigenvalues for matrix-valued Gaussian processes. J. Math. Anal. Appl., 502, 2021 Paper No. 125261, 22.
Song, J., Xiao, Y., Yuan, W., On eigenvalues of the Brownian sheet matrix. 2021 arXiv preprint, arXiv:2103.07378.
Song, J., Yao, J., Yuan, W., Eigenvalue distributions of high-dimensional matrix processes driven by fractional Brownian motion. 2020 arXiv preprint, arXiv:2001.09552.
Song, J., Yao, J., Yuan, W., High-dimensional limits of eigenvalue distributions for general Wishart process. Ann. Appl. Probab. 30 (2020), 1642–1668.
Song, J., Yao, J., Yuan, W., High-dimensional central limit theorems for a class of particle systems. Electron. J. Probab. 26 (2021), 1–33.
Tao, T., Topics in Random Matrix Theory Graduate Studies in Mathematics, vol. 132, 2012, American Mathematical Society, Providence, RI.
Voiculescu, D., Limit laws for random matrices and free products. Invent. Math. 104 (1991), 201–220.
Voiculescu, D.V., Dykema, K.J., Nica, A., Free Random Variables CRM Monograph Series, vol. 1, 1992, American Mathematical Society, Providence, RI.
