Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The gap between probability and prevalence: Loneliness in vector spaces

Author: Maxwell B. Stinchcombe
Journal: Proc. Amer. Math. Soc. 129 (2001), 451-457
MSC (1991): Primary 28C20, 60B11; Secondary 90B40
Published electronically: July 27, 2000
MathSciNet review: 1694881
Full-text PDF

Abstract | References | Similar Articles | Additional Information


The best available definition of a subset of an infinite dimensional, complete, metric vector space $V$ being ``small'' is Christensen's Haar zero sets, equivalently, Hunt, Sauer, and Yorke's shy sets. The complement of a shy set is a prevalent set. There is a gap between prevalence and likelihood. For any probability $\mu$ on $V$, there is a shy set $C$ with $\mu(C) = 1$. Further, when $V$ is locally convex, any i.i.d. sequence with law $\mu$ repeatedly visits neighborhoods of only a shy set of points if the neighborhoods shrink to $0$ at any rate.

References [Enhancements On Off] (What's this?)

  • 1. Anderson, R. M. and W. R. Zame (1997). Genericity with Infinitely Many Parameters. Working Paper, Department of Economics, U. C. Berkeley.
  • 2. Christensen, J. P. R. (1974). Topology and Borel Structure. Amsterdam: North-Holland Publishing Company. MR 50:1221
  • 3. Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth & Brooks/Cole, Pacific Grove, California. MR 91g:60001
  • 4. Hunt, B. R., T. Sauer, and J. A. Yorke (1992). Prevalence: A Translation-Invariant `Almost Every' on Infinite-Dimensional Spaces. Bulletin (New Series) of the American Mathematical Society 27, 217-238. MR 93k:28019
  • 5. Kingman, J. F. C (1967). Additive Set Functions and the Theory of Probability. Proceedings of the Cambridge Philosophical Society 63, 767-775. MR 36:3385
  • 6. Millar, P. W. (1992). Stochastic Search in Banach Spaces. In Probability in Banach Spaces, R. M. Dudley, M. G. Hahn, and J. Kuelbs (eds.), 497-509. MR 94g:62084

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 28C20, 60B11, 90B40

Retrieve articles in all journals with MSC (1991): 28C20, 60B11, 90B40

Additional Information

Maxwell B. Stinchcombe
Affiliation: Department of Economics, University of Texas at Austin, Austin, Texas 78712-1173

Received by editor(s): March 1, 1999
Received by editor(s) in revised form: April 19, 1999
Published electronically: July 27, 2000
Communicated by: Claudia Neuhauser
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society