Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations

Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)

Bai, Z.D., Silverstein, J.: Spectral Analysis of Large Dimensional Random Matrices. Springer Series in Statistics, 2nd edn. Springer, New York (2010)

Ben Arous, G., Guionnet, A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat. Fields 108(4), 517–542 (1997)

Ben Arous, G., Zeitouni, O.: Large deviations from the circular law. ESAIM Probab. Stat. 2, 123–134 (1998)

Bordenave, C., Caputo, P., Chafaï, D.: Spectrum of large random reversible Markov chains: two examples. ALEA Lat. Am. J. Probab. Math. Stat. 7, 41–64 (2010)

Bose, A., Sen, S.: Finite diagonal random matrices. J. Theor. Probab. 26(3), 819–835 (2013)

Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990)

Chafaï, D.: Aspects of large random Markov kernels. Stochastics 81(3–4), 415–429 (2009)

Chung, K.L.: A Course in Probability Theory. Probability and Mathematical Statistics, vol. 21, 2nd edn. Academic, New York (1974)

de Acosta, A., Ney, P.: Large deviations for additive functionals of Markov chains. In: Memoirs of the American Mathematical Society, vol. 228, no. 1070, pp. vi+108 (2014)

Dedecker, J., Merlevède, F., Peligrad, M., Utev, S.: Moderate deviations for stationary sequences of bounded random variables. Ann. Inst. Henri Poincaré Probab. Stat. 45(2), 453–476 (2009)

Dembo, A., Guionnet, A., Zeitouni, O.: Moderate deviations for the spectral measure of certain random matrices. Ann. Inst. H. Poincaré Probab. Stat. 39(6), 1013–1042 (2003)

Dembo, A., Zeitouni, O.: Large deviations techniques and applications. Corrected reprint of the second (1998) edn. In: Stochastic Modelling and Applied Probability, vol. 38, pp. xvi+396. Springer, Berlin (2010)

Diaconis, P., Shahshahani, M.: On the eigenvalues of random matrices. Studies in applied probability. J. Appl. Probab. 31A, 49–62 (1994)

Döring, H., Eichelsbacher, P.: Moderate deviations via cumulants. J. Theor. Probab. 26(2), 360–385 (2013)

Döring, H., Eichelsbacher, P.: Moderate deviations for the eigenvalue counting function of Wigner matrices. ALEA Lat. Am. J. Probab. Math. Stat. 10(1), 27–44 (2013)

Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43(11), 5830–5847 (2002)

Dumitriu, I., Edelman, A.: Global spectrum fluctuations for the $$\beta $$ β -Hermite and $$\beta $$ β -Laguerre ensembles via matrix models. J. Math. Phys. 47(6), 063302 (2006)

Durrett, R.: Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010)

Ellis, R.S.: Large deviations for the empirical measure of a Markov chain with an application to the multivariate empirical measure. Ann. Probab. 16(4), 1496–1508 (1988)

Goldsheid, I.Y., Khoruzhenko, B.A.: The Thouless formula for random non-Hermitian Jacobi matrices. Probability in mathematics. Israel J. Math. 148, 331–346 (2005)

Guionnet, A.: Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Mathematics, vol. 1957. Springer, Berlin (2009)

Han, D., Zhang, D.: Limiting spectral distribution of random birth-death $$Q$$ Q matrices (in Chinese). Sci. Sin. Math. 45, 539–558 (2015). doi: 10.1360/N012015-00065

Hoeffding, W., Robbins, H.: The central limit theorem for dependent random variables. Duke Math. J. 15, 773–780 (1948)

Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)

Lytova, A., Pastur, L.: Central limit theorem for linear eigenvalue statistics of random matrices with independent entries. Ann. Probab. 37(5), 1778–1840 (2009)

Merlevède, F., Peligrad, M.: Moderate deviations for linear processes generated by martingale-like random variables. J. Theor. Probab. 23(1), 277–300 (2010)

Merlevède, F., Peligrad, M., Rio, E.: A Bernstein type inequality and moderate deviations for weakly dependent sequences. Probab. Theory Relat. Fields 151(3–4), 435–474 (2011)

Pastur, L.A., Figotin, A.L.: Spectra of Random and Almost-periodic Operators. Springer, Berlin (1992)

Popescu, I.: General tridiagonal random matrix models, limiting distributions and fluctuations. Probab. Theory Relat. Fields 144(1–2), 179–220 (2009)

Soshnikov, A.: Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields. J. Stat. Phys. 100(3–4), 491–522 (2000)

Soshnikov, A.: Gaussian limit for determinantal random point fields. Ann. Probab. 30(1), 171–187 (2002)

Wu, L.M.: Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23(1), 420–445 (1995)

Zhang, D.: Gaussian fluctuations of eigenvalues in log-gas ensemble: bulk case I. Acta Math. Sin. (Engl. Ser.) 31(9), 1487–1500 (2015)

Zhang, D.: Gaussian fluctuations of eigenvalues in unitary invariant ensemble, arXiv:1409.4008