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Infinite τT products of distribution functions

Part of: Probability theory on algebraic and topological structures Limit theorems

Published online by Cambridge University Press:  09 April 2009

Richard Moynihan
Richard Moynihan
Affiliation:
Analysis Department The MITRE Corporation, Bedford, Massachusetts 01730, U.S.A.
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Abstract

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Let T be a continuous t-norm (a suitable binary operation on[0, 1]) and Δ + the space of distribution functions which are concertratede on [0,∞. theτT product of any F, G in Δ+ is defined at any real x by , and the pair (Δ+, τT) forms a semigroup. Thus, given a sequence {Fi} in Δ+, the n-fold product τT(F1 … Fn) is well-defined for each n. Moreover, that resulting sequence {τT(F1, …, Fn)} is pointwise non-increasing and hence has a weak limit. This paper establishes a convergence theorem which yields a representation for this weak limit. In addition, we prove the Zero-One law that, for Archimedean t-norms, the weak limit is either identically zero or has supremum 1.

MSC classification

Secondary: 60F99: None of the above, but in this section 60B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 2 , September 1978 , pp. 227 - 240
Copyright
Copyright © Australian Mathematical Society 1978

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