For random linear combinations of OPUC in the Nevai class with complex Gaussian coefficients, the variance of the number of zeros in an annulus away from the unit circle converges to an explicit closed-form formula.
Random polynomials: central limit theorems for the real roots
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abstract
The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood-Offord and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova's theorem.
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The Variance of the Number of Zeros for Complex Random Polynomials Spanned by OPUC
For random linear combinations of OPUC in the Nevai class with complex Gaussian coefficients, the variance of the number of zeros in an annulus away from the unit circle converges to an explicit closed-form formula.