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Strong Laws of Large Numbers in an \(F^\alpha \)-Scheme

  • Paul Doukhan
  • Oleg I. Klesov
  • Josef G. Steinebach
Chapter

Abstract

We study the almost sure limiting behavior of record times and the number of records, respectively, in a (so-called) \(F^\alpha \)-scheme. It turns out that there are certain “dualities” between the latter results, that is, under rather general conditions strong laws for record times can be derived from the corresponding ones for the number of records, but in general not vice versa. The results extend, for example, the classical strong laws of Rényi (Annals Faculty Science University Clermont-Ferrand 8:7–12, 1962; Selected Papers of Alfred Rényi, vol. 3, pp. 50–65, Akadémiai Kiadó, Budapest 1976) for record times and counts.

Keywords

Number of records Record times \(F^\alpha \)-scheme Almost sure convergence Strong law of large numbers 

Notes

Acknowledgments

The research of P. Doukhan has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01).

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Paul Doukhan
    • 1
  • Oleg I. Klesov
    • 2
  • Josef G. Steinebach
    • 3
  1. 1.Université Cergy-PontoiseCergy-Pontoise CedexFrance
  2. 2.Department of Mathematical Analysis and Probability TheoryNational Technical University of Ukraine (KPI)KyivUkraine
  3. 3.Mathematisches InstitutUniversität zu KölnCologneGermany

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