Asymptotics of the variance of the number of real roots of random trigonometric polynomials
The average number of real zeros of polynomial \(T_n(\theta)=\sum_{j=1}^{n} a_j \cos j\theta \) where the coefficients \(\{a_j\}_{j=0}^n\) are a sequence of independent identically distributed Gaussian random variables is well-known. This average number, denoted by \(EN_n\) in the internal \((0, 2\pi)\), is asymptotic to \(\operatorname{E}N_n \sim 2n/\sqrt{3} \), for all sufficiently large \(n\). The problem is more complicated when we look for an asymptotic estimate for the variance of the number of real zeros. An upper limit is obtained by the reviewer in [J. Appl. Math. Stochastic Anal. 10, No. 1, 57--66 (1997; Zbl 0880.60058)]. It was acknowledged, since the upper limit obtained was too large, it would not be applicable to obtain an asymptotic estimate for the variance. The present paper successfully re-examined the problem and, using a new approach, obtained an estimate for the variance of the number of real zeros. This paper should pave the way for a study of the distribution of zeros of the above random trigonometric polynomials \(T_n(\theta)\). The latter distribution is known for polynomials of the form \(\sum_{j=1}^{n} (a_j \cos j\theta +b_j \sin j\theta )\), which has the property of being stationary on \(\theta\) and, therefore, the computations are slightly easier. A summary of the earlier results and list of references on this subject is given in [the reviewer, Topics in random polynomials. Harlow: Addison Wesley Longman (1998; Zbl 0949.60010)].
scientific article; zbMATH DE number 6131938
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