Tauberian theory for the asymptotic forms of statistical frequency functions

Abstract
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 functionf(t) is an average of the values off(t) corresponding to individual values oft;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 oft.

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