Variants of Karamata’s iteration theorem

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 nec- essary 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.