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The law of large numbers with exceptional sets

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  • Berkes, István

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.

Suggested Citation

  • Berkes, István, 2001. "The law of large numbers with exceptional sets," Statistics & Probability Letters, Elsevier, vol. 55(4), pages 431-438, December.
  • Handle: RePEc:eee:stapro:v:55:y:2001:i:4:p:431-438
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    References listed on IDEAS

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    1. Fahrner, I. & Stadtmüller, U., 1998. "On almost sure max-limit theorems," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 229-236, March.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
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