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Acta Mathematica Sinica, English Series

, Volume 30, Issue 8, pp 1353–1364 | Cite as

On the laws of large numbers for double arrays of independent random elements in Banach spaces

  • Andrew Rosalsky
  • Le Van Thanh
  • Nguyen Thi Thuy
Article

Abstract

For a double array of independent random elements {V mn ,m ≥ 1, n ≥ 1} in a real separable Banach space, conditions are provided under which the weak and strong laws of large numbers for the double sums Σ i=1 m Σ j=1 n V ij , m ≥ 1, n ≥ 1 are equivalent. Both the identically distributed and the nonidentically distributed cases are treated. In the main theorems, no assumptions are made concerning the geometry of the underlying Banach space. These theorems are applied to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type strong laws of large numbers for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces.

Keywords

Real separable Banach space double array of independent random elements strong and weak laws of large numbers almost sure convergence convergence in probability Rademacher type p Banach space 

MR(2010) Subject Classification

60F05 60F15 60B11 60B12 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Andrew Rosalsky
    • 1
  • Le Van Thanh
    • 2
  • Nguyen Thi Thuy
    • 2
  1. 1.Department of StatisticsUniversity of FloridaGainesvilleUSA
  2. 2.Department of MathematicsVinh UniversityNghe AnVietnam

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