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An estimate of the minimum of the absolute value of trigonometric polynomials with random coefficients
A. G. Karapetyan M. V. Lomonosov Moscow State University
Abstract:
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
$E_k$,
$k\in\mathbb N$,
$0<k<\frac{k_0}{2}$, 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 $\frac{10}{\varepsilon}<r<\frac{11}{\varepsilon}$ 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.
UDC:
517.518 Received: 01.05.1997