Comptes Rendus
Probabilités
Some applications of the Menshov–Rademacher theorem
Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 861-870.

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 n=1 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 (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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.225

Safari Mukeru 1

1 Department of Decision Sciences, University of South Africa, P. O. Box 392, Pretoria, 0003. South Africa
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2021__359_7_861_0,
     author = {Safari Mukeru},
     title = {Some applications of the {Menshov{\textendash}Rademacher} theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {861--870},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {7},
     year = {2021},
     doi = {10.5802/crmath.225},
     zbl = {07398738},
     language = {en},
}
TY  - JOUR
AU  - Safari Mukeru
TI  - Some applications of the Menshov–Rademacher theorem
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 861
EP  - 870
VL  - 359
IS  - 7
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.225
LA  - en
ID  - CRMATH_2021__359_7_861_0
ER  - 
%0 Journal Article
%A Safari Mukeru
%T Some applications of the Menshov–Rademacher theorem
%J Comptes Rendus. Mathématique
%D 2021
%P 861-870
%V 359
%N 7
%I Académie des sciences, Paris
%R 10.5802/crmath.225
%G en
%F CRMATH_2021__359_7_861_0
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/

[1] Witold Bednorz The complete characterization of a.s. convergence of orthogonal series, Ann. Probab., Volume 41 (2013) no. 2, pp. 1055-1071 | MR | Zbl

[2] Rita Guliano Antonini; Yuriy Kozachenko; Andrei Volodin Convergence of series of dependent φ-subgaussian random variables, J. Math. Anal. Appl., Volume 338 (2008) no. 2, pp. 1188-1203 | DOI | MR | Zbl

[3] Kumar Joag-Dev; Frank Proschan Negative association of random variables with applications, Ann. Stat., Volume 11 (1983), pp. 286-295 | MR | Zbl

[4] Jean-Pierre Kahane Some random series of functions, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, 1985 | MR | Zbl

[5] Boris S. Kashin; Artur A. Saakyan Orthogonal series, Translations of Mathematical Monographs, 75, American Mathematical Society, 1989 | MR | Zbl

[6] Mi-Hwa Ko; Tae-Sung Kim; Kwang-Hee Han A note on the almost sure convergence for dependent random variables in Hilbert space, J. Theor. Probab., Volume 22 (2009) no. 2, pp. 506-513 | MR | Zbl

[7] Pertti Mattila Fourier transform and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge University Press, 2015 | Zbl

[8] Przemysław Matuła A note on the almost sure convergence of sums of negatively dependent random variables, Stat. Probab. Lett., Volume 15 (1992) no. 3, pp. 209-2013 | DOI | MR | Zbl

[9] Vladimir A. Mikhailets; Aleksandr A. Murach General forms of the Menshov–Rademacher, Orlicz and Tandori theorems on orthogonal series, Methods Funct. Anal. Topol., Volume 17 (2011) no. 4, pp. 330-340 | MR | Zbl

[10] Safari Mukeru On the convergence of series of dependent random variables, J. Theor. Probab., Volume 34 (2021) no. 3, pp. 1299-1320 | DOI | MR | Zbl

[11] Habib Naderi; Przemysław Matuła; Mahdi Salehi; Mohammad Amini On weak law of large numbers for sums of negatively superadditive dependent random variables, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 1, pp. 13-21 | MR | Zbl

[12] Adam Paszkiewicz The explicit characterization of coefficients of a.e. convergent orthogonal series, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 19-20, pp. 1213-1216 | DOI | MR | Zbl

[13] Adam Paszkiewicz A complete characterization of coefficients of a.e. convergent orthogonal series and majorizing measures, Invent. Math., Volume 180 (2010) no. 1, pp. 55-110 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique