Abstract

We prove several results in the a.s. convergence of weighted sums and strong law of large numbers for convex compact integrable random sets in separable Banach spaces via the norm compactness of the expectation of convex compact integrable random sets and an embedding method. The techniques are applied to prove strong law of large numbers for fuzzy random variables and other related results. Tightness conditions are discussed in comparison with earlier results in the literature.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Expectation
  • fuzzy convex
  • strong law of large numbers
  • tight
  • Toeplitz sequence
  • weighted sum

Mathematics Subject Classification

28B20, 60G42, 46A17, 54A20

References

  1. A. Amrani, C. Castaing, M. Valadier: Méthodes de troncature appliquées à des problèmes de convergence faible ou forte dans L1, Arch. Ration. Mech. Anal. 117 (1992) 167–191.https://doi.org/10.1007/BF00387764Google Scholar
  2. P. Bai, P. Y. Chen, S. H. Sung: On complete convergence and the strong law of large numbers for pairwise independent random variables, Acta Math. Hung. 142(2) (2014) 502–518.https://doi.org/10.1007/s10474-013-0370-4Google Scholar
  3. C. Castaing, C. Godet-Thobie, T. D. Hoang, P. Raynaud de Fitte: On the integration of fuzzy level sets, in: Advances in Mathematical Economics 19, S. Kusuoka, T. Maruyama (eds.), Springer, Tokyo (2015) 1–32.https://doi.org/10.1007/978-4-431-55489-9_1Google Scholar
  4. C. Castaing: Quelques résultats de compacité liés a l’intégration, C. R. Acad. Sci., Paris, Sér. A 270 (1970) 1732–1735; Bull. Soc. Math. Fr., Suppl., Mém. 31–32 (1972) 73–81.https://doi.org/10.1090/S0002-9939-1972-0289606-3Google Scholar
  5. C. Castaing: Weak compactness and convergences in Bochner and Pettis integration, Vietnam J. Math. 24(3) (1996) 241–286.Google Scholar
  6. C. Castaing, Ch. Hess, M. Saadoune: Tightness conditions and integrability of the sequential weak upper limit of a sequence of multifunctions, in: Advances in Mathematical Economics 11, S. Kusuoka, A. Yamazaki (eds.), Springer, Tokyo (2008) 11–44.https://doi.org/10.1007/978-4-431-77784-7_2Google Scholar
  7. C. Castaing, P. Raynaud de Fitte, M. Valadier: Young Measures on Topological Spaces. With Applications in Control Theory and Probability Theory, Kluwer, Dordrecht (2004).https://doi.org/10.1007/1-4020-1964-5Google Scholar
  8. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  9. C. Castaing, P. Raynaud de Fitte: Law of large numbers and Ergodic theorem for convex weak star compact valued Gelfand-integrable mappings, in: Advances in Mathematical Economics 17, S. Kusuoka, T. Maruyama (eds.), Springer, Tokyo (2013) 1–37.https://doi.org/10.1007/978-4-431-54324-4_1Google Scholar
  10. C. Castaing, P. Raynaud de Fitte: U-Uniform scalar integrability and strong laws of large numbers for Pettis-integrable functions with values in a separable locally convex space, J. Theor. Probab. 13(1) (2000) 93–134.https://doi.org/10.1023/A:1007782825974Google Scholar
  11. J. A. Cuesta, C. Matran: Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions, J. Multivariate Anal. 25 (1988) 311–322.https://doi.org/10.1016/0047-259X(88)90054-1Google Scholar
  12. G. Debreu: Integration of correspondences, in: Proc. 5th Berkeley Symp. Math. Stat. Probab. (Berkeley, 1965–1966), Vol. II, Part. 1, L. M. Le Cam, J. Neyman (eds.), University of California, Berkeley (1967) 351–372.Google Scholar
  13. N. Etemadi: An elementary proof of the strong law of large numbers, Z. Wahrscheinlichkeitstheor. Verw. Geb. 55 (1981) 119–122.https://doi.org/10.1007/BF01013465Google Scholar
  14. F. Hiai, H. Umegaki: Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149–182.https://doi.org/10.1016/0047-259X(77)90037-9Google Scholar
  15. H. Inoue: A strong law of large numbers for fuzzy random sets, Fuzzy Sets Syst. 41 (1991) 285–291.https://doi.org/10.1016/0165-0114(91)90132-AGoogle Scholar
  16. S. Y. Joo, Y. K. Kim, J. S. Kwon: Strong convergence for weighted sums of fuzzy random sets, Inf. Sci. 176 (2006) 1086–1099.https://doi.org/10.1016/j.ins.2005.02.002Google Scholar
  17. E. P. Klement, M. L. Puri, D. A. Ralescu: Limit theorems for fuzzy random variables, Proc. R. Soc. Lond., Ser. A 407 (1986) 171–182.https://doi.org/10.1098/rspa.1986.0091Google Scholar
  18. S. Li, Y. Ogura: Strong laws of large numbers for independent fuzzy set-valued random variables, Fuzzy Sets Syst. 157 (2006) 2569–2578.https://doi.org/10.1016/j.fss.2003.06.011Google Scholar
  19. S. Li, Y. Ogura, V. Kreinovich: Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Kluwer, Dordrecht (2002).https://doi.org/10.1007/978-94-015-9932-0Google Scholar
  20. M. L. Puri, D. A. Ralescu: Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409–422.https://doi.org/10.1016/0022-247X(86)90093-4Google Scholar
  21. M. L. Puri, D. A. Ralescu: Strong law of large numbers for Banach space valued random sets, Ann. Probab. 11 (1983) 222–224.https://doi.org/10.1214/aop/1176993671Google Scholar
  22. V. K. Rohatgi: Convergence of weighted sums of independent random variables, Proc. Camb. Philos. Soc. 69 (1971) 305–307.https://doi.org/10.1017/S0305004100046685Google Scholar
  23. R. L. Taylor, H. Inoue: A strong law of large numbers for random sets in Banach spaces, Bull. Inst. Math., Acad. Sin. 13 (1985) 403–409.Google Scholar
  24. R. L. Taylor, H. Inoue: Convergence of weighted sums of random sets, Stochastic Anal. Appl. 3 (1985) 379–396.https://doi.org/10.1080/07362998508809069Google Scholar
  25. R. L. Taylor, D. Wei: Laws of large numbers for tight random elements in normed linear spaces, Ann. Probab. 7 (1979) 150–155.https://doi.org/10.1214/aop/1176995156Google Scholar
  26. M. Valadier: Young measures, in: Methods of Nonconvex Analysis (Varenna, 1989), A. Cellina (ed.), Lecture Notes in Mathematics 1446, Springer, Berlin (1990) 152–188.https://doi.org/10.1007/BFb0084935Google Scholar
  27. X. C. Wang, M. B. Rao: Some results on the convergence of weighted sums of random elements in separable Banach spaces, Stud. Math. 86 (1987) 131–153.https://doi.org/10.4064/sm-86-2-131-153Google Scholar
  28. L. H. Zadeh: Fuzzy sets, Inform. Control 8 (1966) 338–353.https://doi.org/10.1016/S0019-9958(65)90241-XGoogle Scholar

Suggested citation

N. V. Quang, H. T. Duyen. “Convergence of Weighted Sums and Strong Law of Large Numbers for Convex Compact Integrable Random Sets and Fuzzy Random Sets.” Journal of Convex Analysis 24 (2017), No. 1, 213–238. https://doi.org/10.68381/jca24016

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press.