FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1998, VOLUME 4, NUMBER 1, PAGES 245-302
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.
All articles are published in Russian.
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Last modified: April 8, 1998