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Almost sure versions of the Darling-Erdös theorem

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  • Berkes, István
  • Weber, Michel

Abstract

We prove a.s. limit theorems corresponding to the classical Darling-Erdös theorem for the maxima of normalized partial sums of i.i.d. random variables. Our results yield the analogue of the a.s. central limit theorem for the Darling-Erdös max functional and its variants. Unlike in standard a.s. central limit theory, our theorems involve nonlogarithmic averages.

Suggested Citation

  • Berkes, István & Weber, Michel, 2006. "Almost sure versions of the Darling-Erdös theorem," Statistics & Probability Letters, Elsevier, vol. 76(3), pages 280-290, February.
    • Handle: RePEc:eee:stapro:v:76:y:2006:i:3:p:280-290
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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 & Horváth, Lajos, 2001. "The logarithmic average of sample extremes is asymptotically normal," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 77-98, January.
    3. Fahrner, Ingo, 2001. "A strong invariance principle for the logarithmic average of sample maxima," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 317-337, June.
    4. 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.
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