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The law of large numbers with exceptional sets

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  • Berkes, István
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    Abstract

    We investigate the law of large numbers with exceptional n-sets, i.e. when the theorem is required to hold only for almost all n, in the sense of a suitable measure on the integers. We prove the surprising result that in the presence of such exceptional sets, the weak and strong laws of large numbers become equivalent. We also give necessary and sufficient criteria for the validity of such laws.

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    File URL: http://www.sciencedirect.com/science/article/B6V1D-44MWPSH-7/2/d4d3498236592e44a27f402c9608fae4
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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 55 (2001)
    Issue (Month): 4 (December)
    Pages: 431-438

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    Handle: RePEc:eee:stapro:v:55:y:2001:i:4:p:431-438

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    Related research

    Keywords: Law of large numbers Logarithmic density Almost sure central limit theorem;

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    1. Marcus, Michael B. & Rosen, Jay, 1995. "Logarithmic averages for the local times of recurrent random walks and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 59(2), pages 175-184, October.
    2. Ibragimov, Ildar & Lifshits, Mikhail, 1998. "On the convergence of generalized moments in almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 343-351, November.
    3. Berkes, István & Csáki, Endre, 2001. "A universal result in almost sure central limit theory," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 105-134, July.
    4. Fahrner, I. & Stadtmüller, U., 1998. "On almost sure max-limit theorems," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 229-236, March.
    5. Horvath, Lajos & Khoshnevisan, Davar, 1995. "Weight functions and pathwise local central limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 105-123, September.
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