IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v10y1997i3d10.1023_a1022645526197.html
   My bibliography  Save this article

Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

Author

Listed:
  • André Adler
  • Andrew Rosalsky
  • Andrej I. Volodin

Abstract

For a sequence of constants {a n,n≥1}, an array of rowwise independent and stochastically dominated random elements { V nj, j≥1, n≥1} in a real separable Rademacher type p (1≤p≤2) Banach space, and a sequence of positive integer-valued random variables {T n, n≥1}, a general weak law of large numbers of the form $$\sum {_{j = 1}^{T_n } } a_j (V_{nj} - c_{nj} )/b_{[\alpha _n ]} \xrightarrow{P}0$$ is established where {c nj, j≥1, n≥1}, α n → ∞, b n → ∞ are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {V nj, j≥1, n≥1}. Illustrative examples include one wherein the strong law of large numbers fails.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:3:d:10.1023_a:1022645526197
    DOI: 10.1023/A:1022645526197
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1022645526197
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1022645526197?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. 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.
    2. Adler, André & Rosalsky, Andrew & Taylor, Robert L., 1991. "A weak law for normed weighted sums of random elements in rademacher type p banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 37(2), pages 259-268, May.
    3. Andre Adler & Andrew Rosalsky & Robert L. Taylor, 1989. "Strong laws of large numbers for weighted sums of random elements in normed linear spaces," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 12, pages 1-23, January.
    4. André Adler & Andrew Rosalsky, 1991. "On the weak law of large numbers for normed weighted sums of I.I.D. random variables," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 14, pages 1-12, January.
    Full references (including those not matched with items on IDEAS)

    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. 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.
    2. J. Hoffmann-Jørgensen & K-L. Su & R. L. Taylor, 1997. "The Law of Large Numbers and the Ito-Nisio Theorem for Vector Valued Random Fields," Journal of Theoretical Probability, Springer, vol. 10(1), pages 145-183, January.
    3. Yan Wang & Xuejun Wang, 2021. "Complete f-moment convergence for Sung’s type weighted sums and its application to the EV regression models," Statistical Papers, Springer, vol. 62(2), pages 769-793, April.
    4. 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.
    5. Nan Cheng & Chao Lu & Jibing Qi & Xuejun Wang, 2022. "Complete moment convergence for randomly weighted sums of extended negatively dependent random variables with application to semiparametric regression models," Statistical Papers, Springer, vol. 63(2), pages 397-419, April.
    6. Ieva Axt & Roland Fried, 2020. "On variance estimation under shifts in the mean," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(3), pages 417-457, September.
    7. Yan, Ji Gao, 2018. "On Complete Convergence in Marcinkiewicz-Zygmund Type SLLN for END Random Variables and its Applications," IRTG 1792 Discussion Papers 2018-042, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    8. Sookyo Jeong & Hongseok Namkoong, 2020. "Assessing External Validity Over Worst-case Subpopulations," Papers 2007.02411, arXiv.org, revised Feb 2022.
    9. 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).

    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:spr:jotpro:v:10:y:1997:i:3:d:10.1023_a:1022645526197. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.