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Critical Behavior in Almost Sure Central Limit Theory

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  • Siegfried Hörmann

    (Graz University of Technology)

Abstract

Let X 1,X 2,… be i.i.d. random variables with EX 1=0, EX 1 2 =1 and let S k =X 1+⋅⋅⋅+X k . We study the a.s. convergence of the weighted averages $$D_{N}^{-1}\sum_{k=1}^{N}d_{k}I\biggl\{\frac{S_{k}}{\sqrt{k}}\leq x\biggr\},$$ where (d k ) is a positive sequence with D N =∑ k=1 N d k →∞. By the a.s. central limit theorem, the above averages converge a.s. to Φ(x) if d k =1/k (logarithmic averages) but diverge if d k =1 (ordinary averages). Under regularity conditions, we give a fairly complete solution of the problem for what sequences (d k ) the weighted averages above converge, resp. the corresponding LIL and CLT hold. Our results show that logarithmic averaging, despite its prominent role in a.s. central limit theory, is far from optimal and considerably stronger results can be obtained using summation methods near ordinary (Cesàro) summation.

Suggested Citation

  • Siegfried Hörmann, 2007. "Critical Behavior in Almost Sure Central Limit Theory," Journal of Theoretical Probability, Springer, vol. 20(3), pages 613-636, September.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:3:d:10.1007_s10959-007-0080-3
    DOI: 10.1007/s10959-007-0080-3
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    References listed on IDEAS

    as
    1. Lacey, Michael T. & Philipp, Walter, 1990. "A note on the almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 201-205, 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.
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