Statistical Papers

, 51:227 | Cite as

Limit results for ordered uniform spacings

  • Ismihan Bairamov
  • Alexandre Berred
  • Alexei StepanovEmail author
Regular Article


Let Δ k:n  = X k,n  − X k-1,n (k = 1, 2, . . . , n + 1) be the spacings based on uniform order statistics, provided X 0,n  = 0 and X n+1,n  = 1. Obtained from uniform spacings, ordered uniform spacings 0 = Δ0,n  < Δ1,n  < . . . < Δ n+1,n , are discussed in the present paper. Distributional and limit results for them are in the focus of our attention.


Uniform distribution Order statistics Spacings Ordered spacings Limit theorems 

Mathematics Subject Classification (2000)

60G70 62G30 


  1. Abramson L (1966) The distribution of the smallest sample spacing. Ann Math Stat Abstract 37: 1421Google Scholar
  2. Arnold B, Balakrishnan N, Nagaraja H (1992) A first course in order statistics. Wiley, New YorkzbMATHGoogle Scholar
  3. Bairamov IG, Ozkaya N (2000) On the non-parametric test for two sample problem based on spacings. J Appl Stat Sci 10: 57–68MathSciNetGoogle Scholar
  4. Barton DE, David FN (1956) Some notes on ordered random intervals. J Roy Stat Soc B 18: 79–94MathSciNetGoogle Scholar
  5. David HA, Nagaraja HN (2003) Order statistics, 3rd edn. Wiley, New YorkzbMATHGoogle Scholar
  6. Devroye L (1981) Laws of the iterated logarithm for order statistics of uniform spacings. Ann Probab 9: 860–867zbMATHCrossRefMathSciNetGoogle Scholar
  7. Eryilmaz S, Stepanov A (2008) Runs in an ordered sequence of random variables. Metrika 67: 299–313CrossRefMathSciNetGoogle Scholar
  8. Feller W (1967) An introduction to probability theory and its applications, 2nd edn. Wiley, New YorkGoogle Scholar
  9. Hall PG (1984) Limit theorems for sums of general functions of m-spacings. Math Proc Cambridge Philos Soc 96: 517–532zbMATHCrossRefMathSciNetGoogle Scholar
  10. Hu T, Zhuang W (2005) Stochastic properties of p-spacings of generalized order statistics. Probab Eng Inf Sci 19(2): 259–278CrossRefMathSciNetGoogle Scholar
  11. Kotz S, Nadarajah S (2000) Extreme value distributions. Theory and applications. Imperial College Press, LondonzbMATHGoogle Scholar
  12. Kimball BF (1947) Some basic theories for developing tests of fit for the case of nonparametric probability distribution function. Ann Math Stat 18: 540–548zbMATHCrossRefMathSciNetGoogle Scholar
  13. Levy PP (1939) Sur le division d’un segment par des points choisis au hazard. C.R. Acad Sci Paris 208: 147–149Google Scholar
  14. Moran AP (1947) The random division of an interval. J Roy Stat Soc B9: 92–98Google Scholar
  15. Nevzorov V (2001) Records: Mathematical Theory. American Mathematical Society, ProvidenceGoogle Scholar
  16. Pyke R (1965) Spacings (with discussions). J Roy Stat Soc B 27: 395–449MathSciNetGoogle Scholar
  17. Renyi A (1953) On the theory of order statistics. Acta math Hung 4: 191–231zbMATHCrossRefMathSciNetGoogle Scholar
  18. Sherman B (1950) A random variable related to the spacing of sample values. Ann Math Stat 21: 339–361zbMATHCrossRefMathSciNetGoogle Scholar
  19. Weiss L (1959) The limiting joint distribution of the largest and smallest sample spacings. Ann Math Stat 30: 590–593zbMATHCrossRefGoogle Scholar
  20. Weiss L (1969) The joint asymptotic distribution of the k-smallest sample spacings. J Appl Probab 6: 442–448zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Ismihan Bairamov
    • 1
  • Alexandre Berred
    • 2
  • Alexei Stepanov
    • 3
    Email author
  1. 1.Department of MathematicsIzmir University of EconomicsBalcovaTurkey
  2. 2.Faculté des Sciences et TechniquesUniversité du HavreLe Havre CedexFrance
  3. 3.Department of MathematicsKaliningrad State Technical UniversityKaliningradRussia

Personalised recommendations