Variants of Karamata’s iteration theorem

Abstract

Karamata?s Iteration Theorem is used to refine the asymptotic behavior of iterates of a function, under a more restrictive assumption than Karamata?s, but still involving regular variation. A second result gives a necessary and sufficient integral condition for convergence of a series of iterates. Historical background to the idea of regularly varying sequence precedes a short concluding section on attribution of a probabilistic result.