(FUNDAMENTAL AND APPLIED MATHEMATICS)

1998, VOLUME 4, NUMBER 1, PAGES 245-302

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

A. G. Karapetian

Abstract

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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.
\begin{theorem}
For any $\varepsilon\in (0,1)$ and $n>(C(\xi))^{7654/\varepsilon^3}$
$$\mathsf{Pr} \biggl(\min_{x\in\ttt} \biggl| \sum_{j=0}^{n-1}\xi_j \exp(ijx) \biggr| > n^{-\frac{1}{2}+\varepsilon}\biggr) \leq \frac{1}{n^{\varepsilon^2/62}},$$

where $C(\xi)$ is defined in the paper.
\end{theorem}
In the proof of the theorem we use the method of normal degree and establish the estimates for probabilities of events $E_k$, $k\in\mathbb{N}$, $0<k<\frac{k_0}{2}$, and their pairwise intersections. The events $E_k$ are defined by random vectors $X$:
\begin{multline*}
X=(\Re T(x_k),\ldots,\Re (T^{(r-1)}(x_k)/(in)^{r-1}),\\
\Im T(x_k),\ldots,\Im (T^{(r-1)}(x_k)/(in)^{r-1})),
\end{multline*}
where $r$ is chosen as a natural number, such that $\frac{10}{\varepsilon}<r<\frac{11}{\varepsilon}$ 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.

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