Fluctuations in Salem–Zygmund almost sure Central Limit Theorem

We consider the following family of random trigonometric polynomials form Sn(θ):=1 n ∑k=1nakcos(kθ)+bksin(kθ), where variables {ak,bk}k≥1 are i.i.d. and symmetric, defined on a common probability space (Ω,F,P). In seminal paper [SZ54], Salem Zygmund proved that for Rademacher coefficients, P−almost surely all t∈R limn→+∞ 1 2π∫02πeitSn(θ)dθ=e−t2∕2. To best our knowledge, natural question fluctuations in above almost sure limit has not been tackled so far is precisely object this article. Namely, general symmetric coefficients having finite sixth-moment large class continuous test functions ϕ we prove n1 2π∫02πϕ(S n(θ))dθ−∫Rϕ(t)e−t2 2dt 2π→n→∞LawN0,Σϕ2, variance given by Σϕ2:=σϕ2+c2(ϕ)22E[a14]−3. Here, constant σϕ2 explicit corresponds to case Gaussian c2(ϕ) coefficient order 2 decomposition Hermite polynomial basis. It thus turns out universal since they involve kurtosis coefficients. The method develop here robust allows use Wiener chaos expansions proving CLTs functionals general, non-necessarily Gaussian, fields. combines contributions from Malliavin–Stein method, noise stability results, algebraic considerations around Newton sums Newton–Girard formula.