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 Free Access

Abstract | References | Similar Articles | Additional Information

The best available definition of a subset of an infinite dimensional, complete, metric vector space 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 on , there is a shy set with . Further, when is locally convex, any i.i.d. sequence with law repeatedly visits neighborhoods of only a shy set of points if the neighborhoods shrink to at any rate.

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Additional Information

**Maxwell B. Stinchcombe**

Affiliation:
Department of Economics, University of Texas at Austin, Austin, Texas 78712-1173

Email:
maxwell@eco.utexas.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05543-X

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