Published in Proceedings of the Gottingen Academy of Sciences, January 1, 2002. 21 pages. Copyright © 2002 by Theodore P. Hill and Ulrich Krengel
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
Levy’s classical continuity theorem states that if the pointwise limit of a sequence of characteristic functions exists, then the limit function itself is a characteristic function if and only if the limit function satisfies a single universal limit condition (in his case, the limit at zero is one), in which case the underlying measures converge weakly to the probability measure represented by the limit function. It is the purpose of this article to give a number of direct analogs of L´evy’s theorem for other probability-representing functions including moment sequences, maximal moment sequences, mean-residual-life functions, Hardy-Littlewood maximal functions, and failure-rate functions. In each of these cases the single crucial condition on the limit function often relates to conservation of mass or moment, but a general theory encompassing all of these examples is still missing.