Expected number of real roots of random trigonometric polynomials

2016/01/08 by Flasche, Hendrik
#26C10 #30C15 #42A05 #60F99 #60G15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1601.01841

Abstract

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials X_n(t)=u+(1)/(√(n))∑_k=1ⁿ (A_kcos(kt)+B_ksin(kt)), t∈ [0,2π], u∈ℝ whose coefficients A_k, B_k, k∈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]⊆ [0,2π], we prove that lim_n→∞ (𝔼 N_n[a,b])/(n)=(b-a)/(π√(3)) e^-(u2)/(2).

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