On the supremum of random Dirichlet polynomials

• Autorzy: Mikhail Lifshits , Michel Weber
• Języki publikacji: EN

Abstrakty

EN

We study the supremum of some random Dirichlet polynomials $D_{N}(t) = ∑_{n=2}^{N} εₙdₙn^{-σ-it}$, where (εₙ) is a sequence of independent Rademacher random variables, the weights (dₙ) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials $∑_{n∈ 𝓔_{τ}} εₙn^{-σ-it}$, $𝓔_{τ} = {2 ≤ n ≤ N : P⁺(n) ≤ p_{τ}}$, P⁺(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec,
$𝔼 sup_{t∈ ℝ} |∑_{n=2}^{N} εₙn^{-σ-it}| ≈ (N^{1-σ})/(log N)$.
The proofs are entirely based on methods of stochastic processes, in particular the metric entropy method.

Kategorie tematyczne

• 26D05: Inequalities for trigonometric functions and polynomials
• 30B50: Dirichlet series and other series expansions, exponential series
• 60G17: Sample path properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 182
• Numer: 1
• Strony: 41-65

Daty

wydano

2007

Twórcy

autor

Mikhail Lifshits

• Department of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya pl. 2, 198504, Stary Peterhof, Russia

autor

Michel Weber

• Mathématique (IRMA), Université Louis-Pasteur et C.N.R.S., 7 rue René Descartes, 67084 Strasbourg Cedex, France

Identyfikatory

DOI

10.4064/sm182-1-3