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

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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 m i=1 Σ n j=1 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.

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Authors and Affiliations

  1. Department of Statistics, University of Florida, Gainesville, Florida, 32611-8545, USA

    Andrew Rosalsky

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

    Le Van Thanh & Nguyen Thi Thuy

Authors
  1. Andrew Rosalsky
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  2. Le Van Thanh
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  3. Nguyen Thi Thuy
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Correspondence to Andrew Rosalsky.

Additional information

The second author is supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and the Vietnam National Foundation for Sciences and Technology Development NAFOSTED (Grant No. 101.01.2012.13); the third author is supported by NAFOSTED (Grant No. 101.03.2012.17)

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Rosalsky, A., Van Thanh, L. & Thuy, N.T. On the laws of large numbers for double arrays of independent random elements in Banach spaces. Acta. Math. Sin.-English Ser. 30, 1353–1364 (2014). https://doi.org/10.1007/s10114-014-3507-7

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  • DOI: https://doi.org/10.1007/s10114-014-3507-7

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