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Intermediate regular and \(\Pi\) variation. (English) Zbl 0793.26004
A generalization of regular variation is discussed which is intermediate to extended regular variation and \(O\)-regular variation. Analogous to this intermediate regular variation is intermediate \(\Pi\)-variation, a generalization of \(\Pi\)-variation. Paralleling the theories of regular and \(\Pi\)-variation, we demonstrate uniform convergence and representation theorems. We also prove a Karamata theorem and a Tauberian theorem for intermediate regular variation and in so doing we include an interesting extension to the corresponding results for \(O\)-regular variation.
Contained in our proofs is the resolution of a measurability problem extant in other discussions of generalized regular variation.

MSC:
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26A48 Monotonic functions, generalizations
40E05 Tauberian theorems, general
60F05 Central limit and other weak theorems
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