Sur une inégalité de Littlewood–Salem

Abstract

We study uniform estimates for random functions of the form ∑ k=1 N X k f( n k x) where f∈A( T) , { X k } k⩾1 is a sequence of independent complex random variables such that EX k=0 and E|X k| 2<∞ , and { n k } k⩾1 is a strictly increasing sequence of positive integers. (the case of real sequence n k is also studied). Let σ N 2=∑ k=1 N E|X k| 2 . Under some condition on { X k }, we prove that ∫ Ω sup N⩾1 sup x∈ T ∑ k=1 NX kf(n kx) σ N 2 logn N d P<∞. This result generalizes to some extent Salem–Zygmund inequality. We apply this inequality to study the local regularities of random series of Rademacher type ∑ k=1 ∞ a k ε k f( n k x) and of non-stationnary gaussian type ∑ k=1 ∞ a k g k f( n k x).