Some applications of the Menshov–Rademacher theorem

Mukeru, Safari

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

Pour une suite de variables aléatoires réelles ou complexes $\left(X_n\right)$ et une suite de nombres $\left(a_n\right)$, 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.

Given a sequence $\left(X_n\right)$ of real or complex random variables and a sequence of numbers $\left(a_n\right)$, 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.

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