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Convergence of elements in random normed spaces

Published online by Cambridge University Press:  17 April 2009

Robert Lee Taylor
Robert Lee Taylor
Affiliation:
Department of Mathematics and Computer Science, University of South Carolina, Columbia, South Carolina, USA.
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Abstract

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For a random normed space of mappings into a separable normed linear space, convergence of identically distributed elements in i the random norm (norm distribution) is shown to be equivalent to convergence in measure in the weak linear topology. Convergence in measure in each coordinate of a Schauder basis is also shown to be a necessary and sufficient condition for convergence in the random norm topology. These results have laws of large numbers for random elements in separable normed linear spaces as almost immediate corollaries and illustrate some of the recently obtained laws of large numbers for random elements. Similar results are also given for elements which need not have the same norm distributions, and the results are extended to linear metric spaces. Finally, applications of the results to stochastic processes are considered.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1975 , pp. 31 - 47
Copyright
Copyright © Australian Mathematical Society 1975

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