Fluctuations in Salem–Zygmund almost sure Central Limit Theorem

Abstract

We consider the following family of random trigonometric polynomials of the form $S_n(\mathit{\theta }):=\frac{1}{\sqrt{n}}{\sum }_{k=1}^n{a}_{k}cos(k\mathit{\theta })+b_{k}sin(k\mathit{\theta })$, where the variables ${\{a_k,b_k\}}_{k\ge 1}$ are i.i.d. and symmetric, defined on a common probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$. In the seminal paper [SZ54], Salem and Zygmund proved that for Rademacher coefficients, $\mathbb{P}-$almost surely and for all $t\in \mathbb{R}$

$$\underset{n\to +\mathrm{\infty }}{lim}\frac{1}{2\mathrm{\pi }}{\int }_0^{2\mathrm{\pi }}{e}^{it{S}_n(\mathit{\theta })}d\mathit{\theta }=e^{-t^2∕2}.$$

To the best of our knowledge, the natural question of the fluctuations in the above almost sure limit has not been tackled so far and is precisely the object of this article. Namely, for general i.i.d. symmetric random coefficients having a finite sixth-moment and for a large class of continuous test functions ϕ we prove that

$$\sqrt{n}\left(\frac{1}{2\mathrm{\pi }}{\int }_0^{2\mathrm{\pi }}\mathit{\varphi }(S_n(\mathit{\theta }))d\mathit{\theta }-{\int }_{\mathbb{R}}\mathit{\varphi }(t)\frac{e^{-\frac{t^2}{2}}dt}{\sqrt{2\mathrm{\pi }}}\right)\underset{n\to \mathrm{\infty }}{\overset{\text{Law}}{\to }}\mathcal{N}\left(0,{\mathrm{\Sigma }}_{\mathit{\varphi }}^2\right),$$

where the limit variance is given by ${\mathrm{\Sigma }}_{\mathit{\varphi }}^2:={\mathit{\sigma }}_{\mathit{\varphi }}^2+\frac{c_2{(\mathit{\varphi })}^2}{2}\left(\mathbb{E}[a_1^4]-3\right)$. Here, the constant ${\mathit{\sigma }}_{\mathit{\varphi }}^2$ is explicit and corresponds to the limit variance in the case of Gaussian coefficients and $c_2(\mathit{\varphi })$ is the coefficient of order 2 in the decomposition of ϕ in the Hermite polynomial basis. It thus turns out that the fluctuations are Gaussian and not universal since they involve the kurtosis of the random coefficients.

The method we develop here is robust and allows to use Wiener chaos expansions for proving CLTs for functionals of general, non-necessarily Gaussian, random fields. It combines contributions from the Malliavin–Stein method, noise stability results, and algebraic considerations around Newton sums and the Newton–Girard formula.