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Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights

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  • Hu, Tien-Chung
  • Cabrera, Manuel Ordóñez
  • Volodin, Andrei I.

Abstract

Some notions of uniform integrability of an array of random elements in a separable Banach space with respect to an array of random variables are introduced and characterized, in order to obtain weak laws of large numbers for randomly weighted sums. The paper contains results which generalize some previous results for weighted sums with nonrandom weights, and one of them is used to obtain a result of convergence for sums with a random number of addends. Furthermore, a result of almost everywhere convergence of the sequence of certain conditional expectations of the row sums is obtained.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:51:y:2001:i:2:p:155-164
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    References listed on IDEAS

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    1. Cabrera, M. Ordóñez, 1988. "Limit theorems for randomly weighted sums of random elements in normed linear spaces," Journal of Multivariate Analysis, Elsevier, vol. 25(1), pages 139-145, April.
    2. 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.
    3. Gut, Allan, 1992. "The weak law of large numbers for arrays," Statistics & Probability Letters, Elsevier, vol. 14(1), pages 49-52, May.
    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.
    5. Sung, Soo Hak, 1999. "Weak law of large numbers for arrays of random variables," Statistics & Probability Letters, Elsevier, vol. 42(3), pages 293-298, April.
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    Cited by:

    1. Jose Vidal-Sanz & Miguel Delgado, 2004. "Universal consistency of delta estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 791-818, December.
    2. Manuel Ordóñez Cabrera & Andrew Rosalsky & Mehmet Ünver & Andrei Volodin, 2021. "On the concept of B-statistical uniform integrability of weighted sums of random variables and the law of large numbers with mean convergence in the statistical sense," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 83-102, March.
    3. Shen, Xinmei & Lin, Zhengyan, 2008. "Precise large deviations for randomly weighted sums of negatively dependent random variables with consistently varying tails," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3222-3229, December.
    4. 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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