An~estimate of the minimum of the absolute value of trigonometric polynomials with random coefficients

A. G. Karapetyan

Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 1, pp. 245-302.

Résumé

In this paper the random trigonometric polynomial $T(x)=\sum\limits_{j=0}^{n-1}\xi_j\exp (ijx)$ is studied, where $\xi,\xi_j$ are real independent equally distributed random variables with zero mathematical expectations, positive second and finite third absolute moments. Theorem. For any $\varepsilon\in(0,1)$ and $n>(C(\xi))^{7654/\varepsilon^3}$ $$ \mathsf{Pr}\biggl(\min_{x\in\mathbb T}\biggl|\sum_{j=0}^{n-1}\xi_j\exp(ijx) \biggr|>n^{-\frac{1}{2}+\varepsilon}\biggr)\leq \frac{1}{n^{\varepsilon^2/62}}, $$ where $C(\xi)$ is defined in the paper. In the proof of the theorem we use the method of normal degree and establish the estimates for probabilities of events $E_k$, $k\in\mathbb N$, $0$, and their pairwise intersections. The events $E_k$ are defined by random vectors $X$: $$ X=(\operatorname{Re}T(x_k),\ldots,\operatorname{Re}(T^{(r-1)}(x_k)/(in)^{r-1}), \operatorname{Im}T(x_k),\ldots,\operatorname{Im}(T^{(r-1)}(x_k)/(in)^{r-1})), $$ where $r$ is chosen as a natural number, such that $\frac{10}{\varepsilon}$ for given $\varepsilon$ and $x_k=\frac{2\pi k}{k_0}$, where $k_0$ is the greatest prime number, not greater then $n^{1-\frac{\varepsilon}{20}}$. To find these estimates first of all we obtain inequalities for polynomials and by these inequalities we establish the properties of characteristic functions of random vectors $X$ and their pairwise unions.