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The structure of functions satisfying
the law of large numbers
in a class of locally convex spaces


Author: Robert C. Stolz
Journal: Proc. Amer. Math. Soc. 125 (1997), 1215-1220
MSC (1991): Primary 60B12
DOI: https://doi.org/10.1090/S0002-9939-97-03686-1
MathSciNet review: 1363187
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Abstract | References | Similar Articles | Additional Information

Abstract: For each function $f$ that satisfies the law of large numbers with values in a certain class of locally convex spaces with the Radon-Nikodym property the following decomposition holds: $f=f_1+f_2$, where $f_1$ is integrable by seminorm, and $f_2$ is a Pettis integrable function which is scalarly 0.


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  • 1. Beck, A., On the law of large numbers, Ergodic Theory, Proc. Int. Symp. New Orleans 1961, Academic Press, 1963. MR 28:2188
  • 2. Blondia, C., Locally convex spaces with Radon-Nikodym property, Math. Nachr., 114 (1983), 335-341. MR 85i:46003
  • 3. Diestel, J. and Uhl, J.J., Vector Measures, Math. Surveys 15, American Mathematical Society, Providence, RI, 1977. MR 56:12216
  • 4. Dobri\'{c}, V., Analytically normed spaces, Math. Scand. 60 (1987), 109-128. MR 89c:46014
  • 5. -, The law of large numbers in locally convex spaces, Prob. Theor. and Related Fields 78 (1988), 403-417. MR 89e:60065
  • 6. -, The decomposition theorem for functions satisfying the law of large numbers, J. Theor. Prob. 3 (1990), 189-196. MR 91m:60012
  • 7. Grothendieck, A., Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Soc. 16 (1967). MR 17:763c
  • 8. Hoffmann-Jørgensen, J., Stochastic Processes on Polish Spaces, Aarus Univ. Inst. Various Pub. Ser. 39, 1991. MR 95a:60047
  • 9. -, The Theory of Analytic spaces, Aarus Univ. Mat. Inst. Various Pub. Ser. 10, 1970. MR 53:13500
  • 10. -, The law of large numbers for non-measurable and non-separable random elements, Asterisque 131 (1985), 299-356. MR 88a:60014
  • 11. Mourier, E., Elements aleatoires dans un espace de Banach, Ann. Inst. Poincare 13 (1953), 161-299. MR 16:268a
  • 12. Rogers, C. A. and Jayne J. E., K-analytic sets, in Analytic Sets, Academic Press, Inc., London, pp. 2-181, 1980 MR 82m:03063
  • 13. Rudin, W. Functional Analysis, McGraw-Hill Book Co., New York-London, 1973. MR 51:1315
  • 14. Talagrand, M., The Glivenko-Cantelli problem, Ann. Probab. 15 (1987), 837-870. MR 88h:60012

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Additional Information

Robert C. Stolz
Affiliation: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042
Address at time of publication: Division of Science and Mathematics, University of the Virgin Islands, St. Thomas, Virgin Islands 00802
Email: StolzR@lafayette.edu, Robert.Stolz@uvi.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03686-1
Keywords: The law of large numbers, locally convex spaces
Received by editor(s): July 14, 1995
Received by editor(s) in revised form: October 10, 1995
Additional Notes: The present paper is part of the author’s doctoral thesis and was carried out under the supervision of Professor V. Dobrić during a stay at Lehigh University.
Communicated by: Richard T. Durrett
Article copyright: © Copyright 1997 American Mathematical Society

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