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Large deviations for sums
of i.i.d. random compact sets

Author: Raphaël Cerf
Journal: Proc. Amer. Math. Soc. 127 (1999), 2431-2436
MSC (1991): Primary 60D05, 60F10
Published electronically: April 8, 1999
MathSciNet review: 1487361
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets.

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  • 1. Z. Artstein and R. A. Vitale, A strong law of large numbers for random compact sets, Ann. Prob. 3 (1975), 879-882. MR 52:6825
  • 2. J. W. S. Cassels, Measures of the non-convexity of sets and the Shapley-Folkman-Starr theorem, Math. Proc. Camb. Phil. Soc. 78 (1975), 433-436. MR 52:6570
  • 3. N. Cressie, A central limit theorem for random sets, Z. Wahrscheinlichkeitstheor. Verw. Geb. 49 (1979), 37-47. MR 80m:60018
  • 4. J. D. Deuschel and D. W. Stroock, Large deviations, cademic Press, 1989. MR 90h:60026
  • 5. N. Dunford and J. T. Schwartz, Linear operators. Part I: General theory, John Wiley & Sons, 1958. MR 22:8302
  • 6. A. Dembo and O. Zeitouni, Large deviations techniques and applications, Jones and Bartlett publishers, 1993. MR 95a:60034
  • 7. E. Giné, M. G. Hahn and J. Zinn, Limit theorems for random sets: an application of probability in Banach space results, Proc. Fourth Int. Conf. on Prob. Banach spaces, Oberwolfach (1983), 112-135. MR 85d:60019
  • 8. C. Hess, Multivalued strong laws of large numbers in the slice topology. Applications to integrands., Set-Valued Anal. 2 (1994), 183-205. MR 95m:60019
  • 9. L. Hörmander, Sur la fonction d'appui des ensembles convexes dans un espace localement convexe, Arkiv Matematik 3 (1954), 181-186. MR 16:831e
  • 10. N. N. Lyashenko, On limit theorems for sums of independent compact random subsets of Euclidean space, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 85 (1979), 113-128. MR 81k:60014
  • 11. M. L. Puri and D. A. Ralescu, Limit theorems for random compact sets in Banach space, Math. Proc. Camb. Phil. Soc. 97 (1985), 151-158. MR 86c:60024
  • 12. H. Rådström, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169. MR 13:659c
  • 13. W. Rudin, Functional Analysis, McGraw-Hill, 1973. MR 51:1315
  • 14. W. Rudin, Real and complex analysis, McGraw-Hill, 1966. MR 35:1420
  • 15. R. Starr, Quasi-equilibria in markets with non-convex preferences, Econometrica 37 (1969), 25-38.
  • 16. W. Weil, An application of the central limit theorem for Banach-space-valued random variables to the theory of random sets, Z. Wahrscheinlichkeitstheor. Verw. Geb. 60 (1982), 203-208. MR 83h:60010

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Additional Information

Raphaël Cerf
Affiliation: Université Paris Sud, Mathématique, Bâtiment $425$, 91405 Orsay Cedex, France

Keywords: Cram\'{e}r theorem, random sets, large deviations
Received by editor(s): September 10, 1997
Received by editor(s) in revised form: October 27, 1997
Published electronically: April 8, 1999
Communicated by: Stanley Sawyer
Article copyright: © Copyright 1999 American Mathematical Society

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