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Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions

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  • Cuesta, Juan A.
  • Matrán, Carlos

Abstract

The equivalence of sequences of probability measures jointly with the extension of Skorohod's representation theorem due to Blackwell and Dubins is used to obtain strong convergence of weighted sums of random elements in a separable Banach space. Our results include most of the known work on this topic without geometric restrictions on the space. The simple technique developed gives a unified method to extend results on this topic for real random variables to Banach-valued random elements. This technique is also applied to the proof of strong convergence of some statistical functionals.

Suggested Citation

  • Cuesta, Juan A. & Matrán, Carlos, 1988. "Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions," Journal of Multivariate Analysis, Elsevier, vol. 25(2), pages 311-322, May.
    • Handle: RePEc:eee:jmvana:v:25:y:1988:i:2:p:311-322
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    Cited by:

    1. Charles Castaing & Paul Raynaud de Fitte, 2000. "$$\mathfrak{S}$$ -Uniform Scalar Integrability and Strong Laws of Large Numbers for Pettis Integrable Functions with Values in a Separable Locally Convex Space," Journal of Theoretical Probability, Springer, vol. 13(1), pages 93-134, January.
    2. Cabrera, Manuel Ordóñez & Volodin, Andrei I., 2001. "On conditional compactly uniform pth-order integrability of random elements in Banach spaces," Statistics & Probability Letters, Elsevier, vol. 55(3), pages 301-309, December.

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