Random Sets pp 347-360 | Cite as

Laws of Large Numbers for Random Sets

  • Robert L. Taylor
  • Hiroshi Inoue
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 97)


The probabilistic study of geometrical objects has motivated the formulation of a general theory of random sets. Central to the general theory of random sets are questions concerning the convergence for averages of random sets which are known as laws of large numbers. General laws of large numbers for random sets are examined in this paper with emphasis on useful characterizations for possible applications.

Key words

Random Sets Laws of Large Numbers Tightness Moment Conditions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Adler, A. Rosalsky and R.L. TaylorA weak law for normed weighted sums of random elements in Rademacher type p Banach spacesJ. Mult. Anal, 37 (1991), 259–268.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Z. Artstein and R. VitaleA strong law of large numbers for random compact setsAnn. Probab., 3 (1975), 879–882.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Z. Artstein and S. HartLaw of large numbers for random sets and allocation processesMathematics of Operations Research, 6 (1981), 485–492.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Z. Artstein and J.C. HansenConvexification in limit laws of random sets in Banach spacesAnn. Probab., 13 (1985), 307–309.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    N. Cressie, Astrong limit theorem for random setsSuppl. Adv. Appl. Probab., 10 (1978), 36–46.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    N. CressieStatistics for spatial dataWiley, New York (1991).zbMATHGoogle Scholar
  7. [7]
    P.Z. Daffer and R.L. TaylorTightness and strong laws of large numbers in Banach spacesBull. of Inst. Math., Academia Sinica, 10 (1982), 251–263.MathSciNetzbMATHGoogle Scholar
  8. [8]
    G. DebreuIntegration of correspondencesProc. Fifth Berkeley Symp. Math. Statist. Prob., 2 (1966), Univ. of California Press, 351–372.Google Scholar
  9. [9]
    E. Gine, M.G. Hahn and J. ZinnLimit theorems for random sets: An application of probability in Banach space resultsIn Probability in Banach Spaces IV (A. Beck and K. Jacobs, eds.), Lecture Notes in Mathematics, 990, Springer, New York (1983), 112–135.CrossRefGoogle Scholar
  10. [10]
    C. HessTheoreme ergodique et loi forte des grands nombres pour des ensembles aleatoiresComptes Redus de l’Academie des Sciences, 288 (1979), 519–522.MathSciNetzbMATHGoogle Scholar
  11. [11]
    C. HessLoi forte des grands nombres pour des ensembles aleatoires non bornes a valeurs dans un espace de Banach separableComptes Rendus de l’Academie des Sciences, Paris, Serie I (1985), 177–180.Google Scholar
  12. [12]
    J. Hoffmann-Jorgensen and G. PisierThe law of large numbers and the central limit theorem in Banach spacesAnn. Probab., 4 (1976), 587–599.MathSciNetCrossRefGoogle Scholar
  13. [13]
    L. HörmanderSur la fonction d’appui des ensembles convexes dans un espace localement convexeArk. Mat., 3 (1954), 181–186.CrossRefGoogle Scholar
  14. [14]
    H. InoueA limit law for row-wise exchangeable fuzzy random variablesSino-Japan Joint Meeting on Fuzzy Sets and Systems (1990).Google Scholar
  15. [15]
    H. IndueA strong law of large numbers for fuzzy random setsFuzzy Sets and Systems, 41 (1991), 285–291.MathSciNetCrossRefGoogle Scholar
  16. [16]
    H. Inoue and R.L. TaylorA SLLN forarraysof row-wise exchangeable fuzzy random setsStoch. Anal. and Appl., 13 (1995), 461–470.MathSciNetzbMATHGoogle Scholar
  17. [17]
    D.G. KendallFoundations of a theory of random setsIn Stochastic Geometry (ed. E.F. Harding and D.G. Kendall), Wiley (1974), 322–376.Google Scholar
  18. [18]
    E.P. Klement, M.L. Pum and D. RalescuLimit theorems for fuzzy random variablesProc. R. Soc. Lond., A407 (1986), 171–182.Google Scholar
  19. [19]
    R. KruseThe strong law of large numbers for fuzzy random variablesInfo. Sci., 21 (1982), 233–241.MathSciNetCrossRefGoogle Scholar
  20. [20]
    G. MatheronRandom Sets and Integral GeometryWiley (1975).zbMATHGoogle Scholar
  21. [21]
    M. Miyakoshi and M. ShimboA strong law of large numbers for fuzzy random variablesFuzzy Sets and Syst., 12 (1984), 133–142.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    I.S. Molchanov, E. Omey and E. KozarovitzkyAn elementary renewal theorem for random compact convex setsAdv. Appl. Probab., 27 (1995), 931–942.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    E. MourierEléments aleatories dan un espace de BanachAnn. Inst. Henri Poincare, 13 (1953), 159–244.MathSciNetGoogle Scholar
  24. [24]
    M.L. Puri and D.A. RalescuA strong law of large numbers for Banach space-valued random setsAnn. Probab., 11 (1983), 222–224.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    M.L. Puri and D.A. RalescuLimit theorems for random compact sets in Banach spacesMath. Proc. Cambridge Phil. Soc., 97 (1985), 151–158.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    M.L. Puri and D.A. RalescuFuzzy random variablesJ. Math. Anal. ApI., 114 (1986), 409–422.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    H. RadströmAn embedding theorem for spaces of convex setsProc. Amer. Math. Soc., 3 (1952), 165–169.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    H.E. RobbinsOn the measure of a random setAnn. Math. Statist., 14 (1944), 70–74.MathSciNetCrossRefGoogle Scholar
  29. [29]
    H.E. RobbinsOn the measure of a random set IIAnn. Math. Statist., 15 (1945), 342–347.MathSciNetCrossRefGoogle Scholar
  30. [30]
    W.E. Stein and K. TalatiConvex fuzzy random variables FuzzySets and Syst., 6 (1981), 271–283.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    R.L. TaylorStochastic Convergence of Weighted Sums of Random Elements in Linear SpacesLecture Notes in Mathematics, V672 (1978), Springer-Verlag, Berlin-New York.Google Scholar
  32. [32]
    R.L. Taylor, P.Z. Daffer and R.F. PattersonLimit theorems for sums of exchangeable variablesRowman & Allanheld, Totowa N. J. (1985).zbMATHGoogle Scholar
  33. [33]
    R.L. Taylor and H. InoueA strong law of large numbers for random sets in Banach spacesBull. Inst. Math., Academia Sinica, 13 (1985a), 403–409.MathSciNetzbMATHGoogle Scholar
  34. [34]
    R.L. Taylor and H. InoueConvergence of weighted sums of random setsStoch. Anal. and Appl., 3 (1985b), 379–396.MathSciNetzbMATHGoogle Scholar
  35. [35]
    R.L. Taylor and T.-C. HuOn laws of large numbers for exchangeable random variablesStoch. Anal. and Appl., 5 (1987), 323–334.MathSciNetzbMATHGoogle Scholar
  36. [36]
    W. WeilAn application of the central limit theorem for Banach space-valued random variables to the theory of random setsZ. Wharsch. v. Geb., 60 (1982), 203–208.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Robert L. Taylor
    • 1
  • Hiroshi Inoue
    • 2
  1. 1.Department of StatisticsUniversity of GeorgiaAthensUSA
  2. 2.School of ManagementScience University of TokyoJapan

Personalised recommendations