2020/02/04 by Pautrel, Thibault
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2002.01380
We consider random trigonometric polynomials of the form fn(t):=(1)/(√(n)) ∑k=1nak cos(k t)+bk sin(k t), where (ak)k≥ 1 and (bk)k≥ 1 are two independent stationary Gaussian processes with the same correlation function ρ: k ↦ cos(kα), with α≥ 0. We show that the asymptotics of the expected number of real zeros differ from the universal one (2)/(√(3)), holding in the case of independent or weakly dependent coefficients. More precisely, for all ε>0, for all ℓ ∈ (√(2),2], there exists α≥ 0 and n≥ 1 large enough such that |(𝔼[N(fn,[0,2π])])/(n)-ℓ|≤ ε, where \mathcal N(fn,[0,2π]) denotes the number of real zeros of the function fn in the interval [0,2π]. Therefore, this result provides the first example where the expected number of real zeros do not converge as n goes to infinity by exhibiting a whole range of possible limits ranging from √(2) to 2.