On the complete convergence of sequences of random elements in Banach spaces

  • N. V. HuanEmail author


For a sequence \({\{X_{n}, {n \geqslant 1}\}}\) of independent random elements taking values in a Rademacher type p Banach space with the k-th partial sum \({S_{k} (k \geqslant 1)}\), we provide necessary and sufficient conditions for the convergence of \({\sum_{n=1}^{\infty} \frac{1}{n}\,\mathbb{P} ({\rm max}_{1 \leqslant{k}\leqslant{n}} \|S_{k}\| > \varepsilon{n}^{\alpha})}\) and \({\sum_{n=1}^{\infty} \frac{{\rm log} n}{n} \, \mathbb{P} (\max_{1\leqslant{k}\leqslant{n}} \|S_{k}\| > \varepsilon{n}^{\alpha})}\) for every \({\varepsilon > 0}\).

Key words and phrases

complete convergence independent random element Rademacher type p Banach space 

Mathematics Subject Classification

60F15 60B12 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baum L.E., Katz M.: Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108–123 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Erdős P.: On a theorem of Hsu and Robbins. Ann. Math. Statistics 20, 286–291 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Erdős P.: Remark on my paper “On a theorem of Hsu and Robbins”. Ann. Math. Statistics 21, 138 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gan S.: Moment inequalities for B-valued random vectors with applications to the strong limit theorems. Statist. Probab. Lett. 67, 111–119 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gut A.: Complete convergence for arrays. Period. Math. Hungar. 25, 51–75 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hsu P.L., Robbins H.: Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U. S. A. 33, 25–31 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huan N.V.: The Baum–Katz theorem for dependent sequences. Acta Math. Hungar. 151, 162–172 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Institute for Computational Science and TechnologySBI Building, Quang Trung Software CityHo Chi Minh CityVietnam
  2. 2.Department of Mathematics and ApplicationsSaigon UniversityHo Chi Minh CityVietnam

Personalised recommendations