Operations with Higher-Order Types of Asymptotic Variation: Filling Some Gaps

Abstract

In this paper we exhibit some results concerning operations with higher-order types of asymptotic variation, results lacking in the general theory developed in previous papers, namely: 1) we show to what extent the standard elementary factorization of a regularly-varying function holds true for higher-order variation; 2) we exhibit an important class of higher-order regularly-varying functions requiring no restrictions on the indexes when performing multiplication; 3) we get non-obvious results on the types of higher-order variation for linear combinations. In addition, partial results are obtained concerning the type of higher-order variation of the inverse of a regularly-varying function whose index belongs to a set of “exceptional” values.