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Expected number of real roots of random trigonometric polynomials (1601.01841v1)
Published 8 Jan 2016 in math.PR
Abstract: We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials $$ X_n(t)=u+\frac{1}{\sqrt{n}}\sum_{k=1}n (A_k\cos(kt)+B_k\sin(kt)), \quad t\in [0,2\pi],\quad u\in\mathbb{R} $$ whose coefficients $A_k, B_k$, $k\in\mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a, b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]\subseteq [0,2\pi]$, we prove that $$ \lim_{n\rightarrow\infty} \frac{\mathbb{E} N_n[a,b]}{n}=\frac{b-a}{\pi\sqrt{3}} e{-\frac{u2}{2}}. $$