zbMATH — the first resource for mathematics

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.

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
Full Text: DOI