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Domains of Geometric Partial Attraction of Max-Semistable Laws: Structure, Merge and Almost Sure Limit Theorems

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  • Zoltán Megyesi

    (University of Szeged)

Abstract

Max-semistable laws arise as non-degenerate weak limits of suitably centered and normed maxima of i.i.d. random variables along subsequences {k(n)}⊂ℕ such that k(n+1)/k(n)→c≥1, in which case the common distribution function F of the i.i.d. random variables is said to belong to the domain of geometric partial attraction of the max-semistable law. We give a necessary and sufficient condition for F to belong to the domain of geometric partial attraction of a max-semistable law and investigate the structure of these domains. We show that although weak convergence does not take place along {n}=ℕ, the distributions of the maxima “merge” together along the entire {n} with a suitably chosen family of limiting laws. The use of merge is demonstrated by almost sure limit theorems, which are also valid along the whole {n}.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:15:y:2002:i:4:d:10.1023_a:1020692805345
    DOI: 10.1023/A:1020692805345
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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, 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.
    3. Berkes, István & Csáki, Endre & Csörgo, Sándor, 1999. "Almost sure limit theorems for the St. Petersburg game," Statistics & Probability Letters, Elsevier, vol. 45(1), pages 23-30, October.
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    1. István Fazekas & Alexey Chuprunov, 2007. "An Almost Sure Functional Limit Theorem for the Domain of Geometric Partial Attraction of Semistable Laws," Journal of Theoretical Probability, Springer, vol. 20(2), pages 339-353, June.

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