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On the weak laws for arrays of random variables

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  • Sung, Soo Hak
  • Hu, Tien-Chung
  • Volodin, Andrei

Abstract

The convergence in probability of the sequence of sums is obtained, where {un,n[greater-or-equal, slanted]1} and {vn,n[greater-or-equal, slanted]1} are sequences of integers, {Xni,un[less-than-or-equals, slant]i[less-than-or-equals, slant]vn,n[greater-or-equal, slanted]1} are random variables, {cni,un[less-than-or-equals, slant]i[less-than-or-equals, slant]vn,n[greater-or-equal, slanted]1} are constants or conditional expectations, and {bn,n[greater-or-equal, slanted]1} are constants satisfying bn-->[infinity] as n-->[infinity]. The work is proved under a Cesàro-type condition which does not assume the existence of moments of Xni. The current work extends that of Gut (1992, Statist. Probab. Lett. 14, 49-52), Hong and Oh (1995, Statist. Probab. Lett. 22, 52-57), Hong and Lee (1996, Bull. Inst. Math. Acad. Sinica 24, 205-209), and Sung (1998, Statist. Probab. Lett. 38, 10-105).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:72:y:2005:i:4:p:291-298
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Gut, Allan, 1992. "The weak law of large numbers for arrays," Statistics & Probability Letters, Elsevier, vol. 14(1), pages 49-52, May.
    4. Dug Hun Hong & Kwang Sik Oh, 1995. "On the weak law of large numbers for arrays," Statistics & Probability Letters, Elsevier, vol. 22(1), pages 55-57, January.
    5. Sung, Soo Hak, 1998. "Weak law of large numbers for arrays," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 101-105, June.
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    Cited by:

    1. Ankirchner, Stefan & Kruse, Thomas & Urusov, Mikhail, 2017. "WLLN for arrays of nonnegative random variables," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 73-78.

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