Random trigonometric polynomials: universality and non-universality of the variance for the number of real roots

Yen Do, Hoi H. Nguyen, Oanh Nguyen

Published 2019-12-26Version 1

It follows from a recent result by Bally, Caramellino, and Poly that for random trigonometric polynomials with iid coefficients satisfying certain continuousness and having bounded moments of all orders, the variance of the number of roots is asymptotically linear in terms of the expectation; furthermore, the multiplicative constant in this linear relationship depends only on the kurtosis of the common distribution of the polynomial's coefficients. In this note, we show that this is a universal phenomenon under a more general condition, which in particular is satisfied by discrete random variables. Our method gives a fine comparison framework throughout Edgeworth expansion, asymptotic Kac-Rice formula, and a detailed analysis of characteristic functions.