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- W2966684828 abstract "Let ${varphi_k}_{k=0}^infty $ be a sequence of orthonormal polynomials on the unit circle (OPUC) with respect to a probability measure $ mu $. We study the variance of the number of zeros of random linear combinations of the form $$ P_n(z)=sum_{k=0}^{n}eta_kvarphi_k(z), $$ where ${eta_k}_{k=0}^n $ are complex-valued random variables. Under the assumption that the distribution for each $eta_k$ satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when ${varphi_k}$ are regular in the sense of Ullman-Stahl-Totik or are such that the measure of orthogonality $mu$ satisfies $dmu(theta)=w(theta)dtheta$ where $w(theta)=v(theta)prod_{j=1}^J|theta - theta_j|^{alpha_j}$, with $v(theta)geq c>0$, $theta,theta_jin [0,2pi)$, and $alpha_j>0$, we give a quantitative estimate on the the variance of the number of zeros of $P_n$ in sectors that intersect the unit circle. When ${varphi_k}$ are real-valued on the real-line from the Nevai class and ${eta_k}$ are i.i.d.~complex-valued standard Gaussian, we prove a formula for the limiting value of variance of the number of zeros of $P_n$ in annuli that do not contain the unit circle." @default.
- W2966684828 created "2019-08-13" @default.
- W2966684828 creator A5058916861 @default.
- W2966684828 date "2019-08-06" @default.
- W2966684828 modified "2023-09-27" @default.
- W2966684828 title "The Variance of the Number of Zeros for Complex Random Polynomials Spanned by OPUC" @default.
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