On the Asymptotic Distribution of Singular Values of Powers of Random Matrices

Alexeev, N.; Götze, F.; Tikhomirov, A.
May 2014
Journal of Mathematical Sciences;May2014, Vol. 199 Issue 2, p68
Academic Journal
We consider powers of random matrices with independent entries. Let X, i, j ≥ 1, be independent complex random variables with E X = 0 and E| X| = 1, and let X denote an n × n matrix with | X| − X for 1 ≤ i, j ≤ n. Denote by $ s_1^{(m)}\geq \ldots \geq s_n^{(m) } $ the singular values of the random matrix $ \mathbf{W}:={n^{{-\frac{m}{2}}}}{{\mathbf{X}}^m} $ and define the empirical distribution of the squared singular values by where I denotes the indicator of an event B. We prove that the expected spectral distribution $ F_n^{(m) }(x)=\mathbf{E}\mathcal{F}_n^{(m) }(x) $ converges under the Lindeberg condition to the distribution function G( x) defined by its moments


Related Articles

  • On almost sure convergence of the spectral distribution of a power of a random matrix to the Fuss-Catalan distribution. Alexeev, N. // Journal of Mathematical Sciences;Jul2011, Vol. 176 Issue 2, p112 

    In this paper, we considered a power of a non-Hermitian random matrix and prove that the empirical distribution of its singular values converges to the Fuss-Catalan distribution almost surely. Bibliography: 3 titles.

  • 1773. A comparison of two symptom selection methods in vibration-based turbomachinery diagnostics. Gałka, Tomasz // Journal of Vibroengineering;Nov2015, Vol. 17 Issue 7, p3505 

    Complex diagnostic objects, e.g. critical rotating machines, usually generate a large number of diagnostic symptoms. A procedure is therefore required of selecting those most suitable from the point of view of technical condition evolution representation. A method based on the Singular Value...

  • Precise asymptotics for random matrices and random growth models. Su, Zhong Gen // Acta Mathematica Sinica;Jun2008, Vol. 24 Issue 6, p971 

    The author considers the largest eigenvalues of random matrices from Gaussian unitary ensemble and Laguerre unitary ensemble, and the rightmost charge in certain random growth models. We obtain some precise asymptotics results, which are in a sense similar to the precise asymptotics for sums of...

  • Circular law theorem for random Markov matrices. Bordenave, Charles; Caputo, Pietro; Chafaï, Djalil // Probability Theory & Related Fields;Apr2012, Vol. 152 Issue 3/4, p751 

    Let ( X) be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ. Let M be the n × n random Markov matrix with i.i.d. rows defined by $${M_{jk}=X_{jk}/(X_{j1}+\cdots+X_{jn})}$$. In particular, when X follows an exponential law, the random matrix...

  • FROM MATCHBOX TO BOTTLE: A STORAGE PROBLEM. Baringhaus, Ludwig; Grubel, Rudolf // Journal of Applied Probability;Sep2002, Vol. 39 Issue 3, p650 

    Presents a generalization of a matchbox problem concerning the demands of random size where a container is selected at random in successive steps. Case of exponentially distributed demands; Background on asymptotic distribution; Replacement of the matchbox with bottles.

  • Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits. Kuijlaars, Arno; Zhang, Lun // Communications in Mathematical Physics;Dec2014, Vol. 332 Issue 2, p759 

    Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer...

  • Vehicle classification approach based on the combined texture and shape features with a compressive DL. Wei Sun; Xiaorui Zhang; Shunshun Shi; Xiaozheng He // IET Intelligent Transport System;2019, Vol. 13 Issue 7, p1069 

    Automatic vehicle classification is a fundamental task in intelligent transportation systems. Image-based vehicle classification is challenging due to occlusion, low-illumination, and scale change. This study proposes an innovative approach by combining texture and shape features into a...

  • Eigenvalue distributions of random unitary matrices. Wieand, K. // Probability Theory & Related Fields;2002, Vol. 123 Issue 2, p202 

    Let U be an n × n random matrix chosen from Haar measure on the unitary group. For a fixed arc of the unit circle, let X be the number of eigenvalues of M which lie in the specified arc. We study this random variable as the dimension n grows, using the connection between Toeplitz matrices...

  • Linear Statistics of Point Processes via Orthogonal Polynomials. Ryckman, E. // Journal of Statistical Physics;Aug2008, Vol. 132 Issue 3, p473 

    For arbitrary �>0, we use the orthogonal polynomials techniques developed in (Killip and Nenciu in , ; Killip and Nenciu in Int. Math. Res. Not. 50: 2665�2701, ) to study certain linear statistics associated with the circular and Jacobi � ensembles. We identify the distribution of...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics