(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 $T\left(x\right)= \sum$j=0n-1 ξ jexp(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 $\epsilon \in \left(0,1\right)$ and $n>\left(C\left(\xi \right)\right)^\left\{7654/\varepsilon ^3\right\}$

$\mathsf\left\{Pr\right\} \biggl \left(\min _\left\{x\in \ttt\right\}\biggl | \sum _\left\{j=0\right\}^\left\{n-1\right\} \xi _j \exp \left(ijx\right) \biggr| > n^\left\{-\frac\left\{1\right\}\left\{2\right\}+\varepsilon \right\}\biggr \right)\leq \frac \left\{1\right\}\left\{n^\left\{\varepsilon ^2/62\right\}\right\},$

where $C\left( \xi \right)$ 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, $k \in$N, $0 < k < \left(k$0)/2, and their pairwise intersections. The events $E$k are defined by random vectors $X$:

$X=\left(\Re T\left(x k\right),\ldots ,\Re \left(T \left(r-1\right)\left(x k\right)/\left(in\right) r-1\right), \Im T\left(x k\right),\ldots ,\Im \left(T \left(r-1\right)\left(x k\right)/\left(in\right) r-1\right)\right),$

where $r$ is chosen as a natural number, such that $10/\left(\epsilon \right) < r < 11/\left(\epsilon \right)$ for given $\epsilon$ and $x$k=(2 π k)/(k0), where $k$0 is the greatest prime number, not greater then n1-(ε)/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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