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


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In this paper the random trigonometric polynomial T(x)= ∑ j=0n-1 ξ jexp(ijx) is studied, where ξ , ξ j are real independent equally distributed random variables with zero mathematical expectations, positive second and finite third absolute moments.

Theorem. For any ε ∈ (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( ξ ) 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 Ek, k ∈ N, 0 < k < (k0)/2, and their pairwise intersections. The events Ek are defined by random vectors X:

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

where r is chosen as a natural number, such that 10/(ε) < r < 11/(ε) for given ε and xk=(2 π k)/(k0), where k0 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.

All articles are published in Russian.

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Last modified: April 8, 1998