On some properties of asymptotic quasi-inverse functions

Abstract

We introduce the notions of asymptotic quasi-inverse functions and as-ymptotic inverse functions as weaker versions of (quasi-)inverse functions, and study their main properties. Asymptotic quasi-inverse functions exist in the class of so-called pseudo-regularly varying (PRV) functions, i.e. functions preserving the as-ymptotic equivalence of functions and sequences. On the other hand, asymptotic inverse functions exist in the class of so-called POV functions, i.e., functions with positive order of variation. In this paper, we obtain some new results about PRV and POV functions. Some further properties of asymptotic (quasi-)inverse functions as well as some applications will be discussed in Part II of this paper to appear in no. 71 of this journal.

Some properties of asymptotic quasi-inverse functions and their applications I

“…The paper is divided into two parts, Part I containing Sections 1-7, and Part II (see [7]) containing Sections 8-11.…”

On some extensions of Karamata’s theory and their applications

“…The third part of this paper (Sections 4-6) deals with some properties of functions preserving the equivalence of functions, that is with functions f satisfying f (u(t))/f (v(t)) → 1, whenever u(t)/v(t) → 1 (as t → ∞), and with asymptotic quasi-inverse functions (confer Buldygin et al [9], [15]- [17]; proofs can be found in [11] and [15]- [17]). This part is organized as follows.…”

“…PRV functions and their various applications have been studied by Korenblyum [36], Matuszewska [38], Matuszewska and Orlicz [39], Stadtmüller and Trautner [48], [49], Berman [4,5], Yakymiv [53,54], Cline [19], Klesov et al [33], Djurčić and Torgašev [21], Buldygin et al [9], [11]- [13], [15]- [18]. Note that PRV functions are called regularly oscillating in Berman [4], weakly oscillating in Yakymiv [53] and intermediate regularly varying in Cline [19].…”

“…Moreover, the asymptotic stability with respect to initial conditions of solutions of the above stochastic differential equation as well as the asymptotic behavior of generalized renewal processes connected with this equation are considered in this part. This section is based on Buldygin et al [16]- [18] (for the proofs we refer to Buldygin et al [17]). …”

“…Further applications to the asymptotic behavior of generalized renewal sample functions of continuous functions and sequences were considered in Buldygin et al [11,16].…”