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On complete convergence in mean for double sums of independent random elements in Banach spaces

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Abstract

Conditions are provided under which a normed double sum of independent random elements in a real separable Rademacher type p Banach space converges completely to 0 in mean of order p. These conditions for the complete convergence in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. In case the Banach space is not of Rademacher type p, it is proved that the complete convergence in mean of order p of a normed double sum implies a strong law of large numbers.

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Authors and Affiliations

  1. Department of Mathematics, Vinh University, Vinh, Nghe An, 42118, Vietnam

    L. V. Thanh & N. T. Thuy

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  1. L. V. Thanh
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  2. N. T. Thuy
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Correspondence to L. V. Thanh.

Additional information

The research was supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and Ministry of Education and Training, grant no. B2016-TDV-06.

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Thanh, L.V., Thuy, N.T. On complete convergence in mean for double sums of independent random elements in Banach spaces. Acta Math. Hungar. 150, 456–471 (2016). https://doi.org/10.1007/s10474-016-0665-3

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  • DOI: https://doi.org/10.1007/s10474-016-0665-3

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