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Convergence of weighted sums of tight random elements

Author

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  • Wei, Duan
  • Taylor, R. L.

Abstract

Convergence of weighted sums of tight random elements {Vn} (in a separable Banach space) which have zero expected values and uniformly bounded rth moments (r > 1) is obtained. In particular, if {ank} is a Toeplitz sequence of real numbers, then [Sigma]k=1[infinity] ankf(Vk) --> 0 in probability for each continuous linear functional f if and only if ||[Sigma]k=1[infinity] ankVk ||--> 0 in probability. When the random elements are independent and max1 0 with probability 1. These results yield laws of large numbers without assuming geometric conditions on the Banach space. Finally, these results can be extended to random elements in certain Fréchet spaces.

Suggested Citation

  • Wei, Duan & Taylor, R. L., 1978. "Convergence of weighted sums of tight random elements," Journal of Multivariate Analysis, Elsevier, vol. 8(2), pages 282-294, June.
    • Handle: RePEc:eee:jmvana:v:8:y:1978:i:2:p:282-294
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    Citations

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    Cited by:

    1. André Adler & Andrew Rosalsky & Andrej I. Volodin, 1997. "Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces," Journal of Theoretical Probability, Springer, vol. 10(3), pages 605-623, July.
    2. Rosalsky, Andrew & Thành, Lê Vǎn, 2021. "A note on the stochastic domination condition and uniform integrability with applications to the strong law of large numbers," Statistics & Probability Letters, Elsevier, vol. 178(C).
    3. Hu, Tien-Chung & Cabrera, Manuel Ordóñez & Volodin, Andrei I., 2001. "Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 155-164, January.
    4. Rosalsky, Andrew & Sreehari, M., 1998. "On the limiting behavior of randomly weighted partial sums," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 403-410, November.

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