Abstract
Given a sequence (X n ) of real or complex random variables and a sequence of numbers (a n ), an interesting problem is to determine the conditions under which the series ∑ n=1 ∞ a n X n is almost surely convergent. This paper extends the classical Menshov–Rademacher theorem on the convergence of orthogonal series to general series of dependent random variables and derives interesting sufficient conditions for the almost everywhere convergence of trigonometric series with respect to singular measures whose Fourier transform decays to 0 at infinity with positive rate.
Highlights
A classical fundamental result obtained independently by Menshov and Rademacher in the1920’s states that if the condition ∞ n=1 |an |2 log22 (n + 1) < ∞is satisfied for a given sequence of real or complex numbers, for any sequence of orthonormal sequence of
Kim and Antonini, Kozachenko and Volodin [2] considered the case of sub-Gaussian random variables (Xn) exhibiting certain dependence structures and derived interesting sufficient conditions for the a.s. convergence
Our approach consists of extending the Menshov–Rademacher theorem from the classical case of orthonormal series to general dependent random variables
Summary
A classical fundamental result obtained independently by Menshov and Rademacher in the. Kozachenko and Volodin [2] considered the case of sub-Gaussian random variables (Xn) exhibiting certain dependence structures (negative association, m-dependence and m-acceptability) and derived interesting sufficient conditions for the a.s. convergence. It is shown in [10] that if (Xn) can be expressed as linear combinations Xn =. Rademacher theorem from the classical case of orthonormal series to general dependent random variables. We obtain an explicit sufficient condition for the almost sure convergence of random series in their most generality This condition is applied to study almost everywhere convergence of trigonometric Fourier series on some subsets of the unit circle of Lebesgue measure zero. The paper concludes with a remark concerning the particular case of Gaussian random variables were we prove (using the classical Sudakov–Fernique inequality) that the condition
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