Variations on Salem–Zygmund results for random trigonometric polynomials: application to almost sure nodal asymptotics

Abstract

On some probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, we consider two independent sequences ${(a_k)}_{k\ge 1}$ and ${(b_k)}_{k\ge 1}$ of i.i.d. random variables that are centered with unit variance and which admit a moment strictly higher than two. We then consider the associated random trigonometric polynomial $f_n(t):=\frac{1}{\sqrt{n}}{\sum }_{k=1}^n{a}_{k}cos(kt)+b_{k}sin(kt)$, $t\in \mathbb{R}$. In their seminal work, for Rademacher coefficients, Salem and Zygmund showed that $\mathbb{P}$ almost surely:

$$\forall t\in \mathbb{R},\frac{1}{2\mathrm{\pi }}{\int }_0^{2\mathrm{\pi }}exp\left(it{f}_n(x)\right)dx\underset{n\to \mathrm{\infty }}{\overset{}{\to }}{e}^{-\frac{t^2}{2}}.$$

In other words, if X denotes an independent random variable uniformly distributed over $[0,2\mathrm{\pi }]$, $\mathbb{P}$ almost surely, under the law of X, $f_n(X)$ converges in distribution to a standard Gaussian variable. In this paper, we revisit the above Salem–Zygmund result from different perspectives. Namely,

1. we establish a convergence rate for some adequate metric via the Stein’s method,

2. we prove a functional counterpart of Salem–Zygmund CLT,

3. we extend it to more general distributions for X,

4. we also prove that the convergence actually holds in total variation.

As an application, in the case where the random coefficients have a symmetric distribution and admit a moment of order 4, we show that, $\mathbb{P}$ almost surely, for any interval $[a,b]\subset [0,2\mathrm{\pi }]$, the number of real zeros $\mathcal{N}(f_n,[a,b])$ of $f_n$ in the interval $[a,b]$ satisfies the universal asymptotics

$$\frac{\mathcal{N}(f_n,[a,b])}{n}\underset{n\to +\mathrm{\infty }}{\overset{}{\to }}\frac{(b-a)}{\mathrm{\pi }\sqrt{3}}.$$