FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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 $T(x)=\; \sum$_j=0^n-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 $\epsilon \; \in \; (0,1)$ and $\$n>(C(\backslash xi\; ))^\{7654/\backslash varepsilon\; ^3\}\$$

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, $k\; \in$N, ₀)/2, and their pairwise intersections. The events $E$_k are defined by random vectors $X$:

where $r$ is chosen as a natural number, such that for given $\epsilon$ and $x$_k=(2 π k)/(k₀), where $k$₀ is the greatest prime number, not greater then $$n^1-(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.

All articles are published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/98/981/98120h.htm
Last modified: April 8, 1998