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Spectral measure of large random Helson matrices
Published 29 May 2024 in math.PR and math.NT | (2405.18796v3)
Abstract: We study the limiting spectral measure of large random Helson matrices and large random matrices of certain patterned structures. Given a real random variable $X \in L{2+ \varepsilon}(\mathbb{P}) $ for some $\varepsilon > 0$ and $\mathrm{Var}(X) = 1$. For the random $n \times n$ Helson matrices generated by the independent copies of $X$, scaling the eigenvalues by $\sqrt{n}$, we prove the almost sure weak convergence of the spectral measure to the standard Wigner semi-circular law. Similar results are established for large random matrices with certain general patterned structures.
- An introduction to random matrices, volume 118 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010.
- Tom M. Apostol. Introduction to analytic number theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976.
- Zhidong Bai. Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica, 9(3):611–677, 1999. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author.
- Spectral analysis of large dimensional random matrices. Springer Series in Statistics. Springer, New York, second edition, 2010.
- Matrix concentration inequalities and free probability. Invent. Math., 234(1):419–487, 2023.
- Arup Bose. Random Matrices and Non-Commutative Probability. Chapman and Hall/CRC, 2021.
- Another look at the moment method for large dimensional random matrices. Electron. J. Probab., 13:no. 21, 588–628, 2008.
- Projecting onto Helson matrices in Schatten classes. Studia Math., 256(1):109–119, 2021.
- Failure of Nehari’s theorem for multiplicative Hankel forms in Schatten classes. Studia Math., 228(2):101–108, 2015.
- Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab., 34(1):1–38, 2006.
- Richard M. Dudley. Real analysis and probability, volume 74 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original.
- Proof methods in random matrix theory. Probab. Surv., 20:291–381, 2023.
- Kevin Ford. The distribution of integers with a divisor in a given interval. Ann. of Math. (2), 168(2):367–433, 2008.
- Henry Helson. Hankel forms and sums of random variables. Studia Math., 176(1):85–92, 2006.
- Viktor B. LidskiÄ. On the characteristic numbers of the sum and product of symmetric matrices. Doklady Akad. Nauk SSSR (N.S.), 75:769–772, 1950.
- A helson matrix with explicit eigenvalue asymptotics. Journal of Functional Analysis, 275(4):967–987, 2018.
- A lower bound in Nehari’s theorem on the polydisc. J. Anal. Math., 118(1):339–342, 2012.
- On helson matrices: moment problems, non-negativity, boundedness, and finite rank. Proceedings of the London Mathematical Society, 116(1):101–134, 2018.
- Henry Teicher. Almost certain convergence in double arrays. Z. Wahrsch. Verw. Gebiete, 69(3):331–345, 1985.
- Eugene P. Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2), 67:325–327, 1958.
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