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Random orthonormal polynomials: Local universality and expected number of real roots.
- Source :
- Transactions of the American Mathematical Society; Sep2023, Vol. 376 Issue 9, p6215-6243, 29p
- Publication Year :
- 2023
-
Abstract
- We consider random orthonormal polynomials \begin{equation*} F_{n}(x)=\sum _{i=0}^{n}\xi _{i}p_{i}(x), \end{equation*} where \xi _{0}, ..., \xi _{n} are independent random variables with zero mean, unit variance and uniformly bounded (2+\varepsilon) moments, and (p_n)_{n=0}^{\infty } is the system of orthonormal polynomials with respect to a fixed compactly supported measure on the real line. Under mild technical assumptions satisfied by many classes of classical polynomial systems, we establish universality for the leading asymptotics of the average number of real roots of F_n, both globally and locally. Prior to this paper, these results were known only for random orthonormal polynomials with Gaussian coefficients (see D. D. Lubinsky, I. E. Pritsker, and X. Xie [Proc. Amer. Math. Soc. 144 (2016), pp. 1631–1642]) using the Kac-Rice formula, a method that does not extend to the generality of our paper. [ABSTRACT FROM AUTHOR]
- Subjects :
- POLYNOMIALS
RANDOM variables
INDEPENDENT variables
RANDOM graphs
Subjects
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 376
- Issue :
- 9
- Database :
- Complementary Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 170039158
- Full Text :
- https://doi.org/10.1090/tran/8901