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Extensions of the Borel–Cantelli lemma in general measure spaces

Author

Listed:
  • Xuejun Wang

    (Anhui University)

  • Xinghui Wang

    (Anhui University)

  • Xiaoqin Li

    (Anhui University)

  • Shuhe Hu

    (Anhui University)

Abstract

In this paper, an important bilateral inequality for a sequence of nonnegative measurable functions on a measure space $$(S,\mathcal {B}_S,\mu )$$ ( S , B S , μ ) is obtained, and some sufficient conditions for $$\mu \left( \limsup \limits _{n\rightarrow \infty }A_n\right) =\mu (S)$$ μ lim sup n → ∞ A n = μ ( S ) are given. In addition, a weighted version of the Borel–Cantelli Lemma on the measure space is obtained. Our results generalize the corresponding ones for bounded random sequences to the case of unbounded measurable functions.

Suggested Citation

  • Xuejun Wang & Xinghui Wang & Xiaoqin Li & Shuhe Hu, 2014. "Extensions of the Borel–Cantelli lemma in general measure spaces," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1229-1248, December.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:4:d:10.1007_s10959-013-0526-8
    DOI: 10.1007/s10959-013-0526-8
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    References listed on IDEAS

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    1. Xie, Yuquan, 2009. "A bilateral inequality on a nonnegative bounded random sequence," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1577-1580, July.
    2. B. Prakasa Rao, 2009. "Conditional independence, conditional mixing and conditional association," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(2), pages 441-460, June.
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