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Tauberian theory for the asymptotic forms of statistical frequency functions

Published online by Cambridge University Press:  24 October 2008

J. M. Hammersley
J. M. Hammersley
Affiliation:
Lectureship in the Design and Analysis of Scientific ExperimentUniversity of Oxford
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Extract

An Abelian theorem is a theorem stating that a given behaviour on the part of each of several quantities entails similar behaviour for their average. A Tauberian theorem is a converse to an Abelian theorem. As a rule, a given behaviour of an average will not entail similar behaviour of the individual quantities themselves unless there is some condition imposed to secure reasonably uniform behaviour amongst the individuals. Such a condition, known as a Tauberian condition, is usually sufficient but not necessary, and it enters into the premises of the Tauberian theorem. We interpret ‘average’ in a wide sense to include any kind of smoothing process; for example, the integral of a function f(t) is an average of the values of f(t) corresponding to individual values of t; and we may seek a sufficient Tauberian condition such that a limit-behaviour of an integral-average

entails the corresponding limit-behaviour

for individual values of t.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 4 , October 1952 , pp. 592 - 599
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

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