IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v51y2001i2p155-164.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(00)00146-2
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. Clément de Chaisemartin & Luc Behaghel, 2020. "Estimating the Effect of Treatments Allocated by Randomized Waiting Lists," Econometrica, Econometric Society, vol. 88(4), pages 1453-1477, July.
    4. 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.
    5. Adler, André & Rosalsky, Andrew & Volodin, Andrej I., 1997. "A mean convergence theorem and weak law for arrays of random elements in martingale type p Banach spaces," Statistics & Probability Letters, Elsevier, vol. 32(2), pages 167-174, March.
    6. Pingyan, Chen & Shixin, Gan, 2007. "Limiting behavior of weighted sums of i.i.d. random variables," Statistics & Probability Letters, Elsevier, vol. 77(16), pages 1589-1599, October.
    7. 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.
    8. Hong, Dug Hun & Cabrera, Manuel Ordóñez & Sung, Soo Hak & Volodin, Andrei I., 2000. "On the weak law for randomly indexed partial sums for arrays of random elements in martingale type p Banach spaces," Statistics & Probability Letters, Elsevier, vol. 46(2), pages 177-185, January.
    9. 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.
    10. Arvanitis, Stelios & Louka, Alexandros, 2016. "A CLT for martingale transforms with infinite variance," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 116-123.
    11. 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).
    12. Csörgő, Miklós & Nasari, Masoud M., 2013. "Asymptotics of randomly weighted u- and v-statistics: Application to bootstrap," Journal of Multivariate Analysis, Elsevier, vol. 121(C), pages 176-192.
    13. Cl'ement de Chaisemartin & Diego Ciccia Xavier D'Haultf{oe}uille & Felix Knau, 2024. "Two-way Fixed Effects and Differences-in-Differences Estimators in Heterogeneous Adoption Designs," Papers 2405.04465, arXiv.org, revised Aug 2024.
    14. A. Elijio & V. Čekanavičius, 2015. "Compound Poisson approximation to weighted sums of symmetric discrete variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(1), pages 195-210, February.
    15. Roozegar, Rasool & Soltani, A.R., 2015. "On the asymptotic behavior of randomly weighted averages," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 269-272.
    16. Sung, Soo Hak, 1998. "Weak law of large numbers for arrays," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 101-105, June.
    17. Gut, Allan & Stadtmüller, Ulrich, 2017. "Strong laws for sequences in the vicinity of the LIL," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 63-72.
    18. Hong, Dug Hun, 1996. "On the weak law of large numbers for randomly indexed partial sums for arrays," Statistics & Probability Letters, Elsevier, vol. 28(2), pages 127-130, June.
    19. Christofides, Tasos C. & Serfling, Robert, 1998. "U-statistics on a lattice of I.I.D. random variables," Statistics & Probability Letters, Elsevier, vol. 40(3), pages 293-303, October.
    20. Sung, Soo Hak & Hu, Tien-Chung & Volodin, Andrei, 2005. "On the weak laws for arrays of random variables," Statistics & Probability Letters, Elsevier, vol. 72(4), pages 291-298, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:51:y:2001:i:2:p:155-164. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.