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

  • N. V. HuanEmail author
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Abstract

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 

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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

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