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On the pointwise central limit theorem and mixtures of stable distributions

Author

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  • Berkes, I.
  • Csáki, E.

Abstract

Let X1, X2, ... be i.i.d. r.v.'s with EX1 = 0. EX21 - 1 and put Sn = X1 + ... + Xn. We investigate the a.s. limiting behavior of for general norming sequences (ak). The pointwise central limit theorem shows that LN converges a.s. to the normal distribution if ak = [radical sign]k; in our paper we prove the surprising result that for suitably chosen (ak) the expression LN can converge also to non-Gaussian limits, in particular, any symmetric stable distribution is a possible limit of LN. We shall determine the class of limit distributions of LN and extend the result to the case when Xn belong to the domain of attraction of a stable law.

Suggested Citation

  • Berkes, I. & Csáki, E., 1996. "On the pointwise central limit theorem and mixtures of stable distributions," Statistics & Probability Letters, Elsevier, vol. 29(4), pages 361-368, September.
  • Handle: RePEc:eee:stapro:v:29:y:1996:i:4:p:361-368
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    Cited by:

    1. Zoltán Megyesi, 2002. "Domains of Geometric Partial Attraction of Max-Semistable Laws: Structure, Merge and Almost Sure Limit Theorems," Journal of Theoretical Probability, Springer, vol. 15(4), pages 973-1005, October.

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