Some applications of the Menshov–Rademacher theorem

Safari Mukeru

doi:10.5802/crmath.225

Probability theory

Some applications of the Menshov–Rademacher theorem

Safari Mukeru ¹

¹ Department of Decision Sciences, University of South Africa, P. O. Box 392, Pretoria, 0003. South Africa

Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 861-870.

Abstract
Résumé

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 $\sum_{n = 1}^{\infty} a_{n} X_{n}$ 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.

Pour une suite de variables aléatoires réelles ou complexes $(X_{n})$ et une suite de nombres $(a_{n})$ , une question importante est de savoir sous quelles conditions la série aléatoire $\sum_{n = 1}^{\infty} a_{n} X_{n}$ est convergente presque sûrement. Cette note généralise le théorème classique de Menshov–Rademacher sur la convergence de séries orthogonales aux séries plus générales de variables aléatoires dépendantes et en déduit des conditions suffisantes pour la convergence presque sûre des séries trigonométriques par rapport à des mesures singulières dont la transformée de Fourier tend vers $0$ à l’infini avec un taux positif.

Received: 2021-02-15
Revised: 2021-05-01
Accepted: 2021-05-21
Published online: 2021-09-17

Zbl

DOI: 10.5802/crmath.225

Author's affiliations:

Safari Mukeru ¹

¹ Department of Decision Sciences, University of South Africa, P. O. Box 392, Pretoria, 0003. South Africa

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Some applications of the {Menshov{\textendash}Rademacher} theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {861--870},
     publisher = {Acad\'emie des sciences, Paris},
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Safari Mukeru. Some applications of the Menshov–Rademacher theorem. Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 861-870. doi : 10.5802/crmath.225. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.225/

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