On some properties of asymptotic quasi-inverse functions. (Ukrainian, English) Zbl 1199.26009

Buldygin, V. V.; Klesov, O. I.; Steinebach, J. G.

Teor. Jmovirn. Mat. Stat. 77, 13-27 (2007); translation in Theory Probab. Math. Stat. 77, 15-30 (2008).

A function \(\tilde{f}\) is called asymptotically quasi inverse (a.q.i.) to \(f: \mathbb R\to\mathbb R\) if \(f(\tilde{f}(t))\sim t\) and \( \tilde{f}(t)\to\infty\) as \(t\to\infty\) and asymptotically inverse if it is a.q.i. and \(\tilde{f}(f(t))\sim t\) as \(t\to\infty\). The authors continue studying properties of functions, which are asymptotically (quasi-)inverse to different classes of functions (regularly varying (RV), O-regularly varying (ORV), O-slowly varying (OSV), pseudo-regularly varying functions (PRV), sufficiently quickly increasing (SQI), positively increasing). [For the previous results, see Teor. Jmovirn. Mat. Stat. 70, 9–25 (2004); translation in Theory Probab. Math. Stat. 70, 11–28 (2005; Zbl 1103.26001); and Teor. Jmovirn. Mat. Stat. 71, 33–48 (2004); translation in Theory Probab. Math. Stat. 71, 37–52 (2005; Zbl 1101.26003)]. In this article, a characterization of normalizing functions connected with the limiting behaviour of ratios of asymptotic quasi-inverse functions is discussed. For nondecreasing functions, the necessary and sufficient conditions are obtained under which the asymptotically quasi-inverse functions belong to special classes of O-regularly varying functions.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

26A12	Rate of growth of functions, orders of infinity, slowly varying functions
60F99	Limit theorems in probability theory
26A48	Monotonic functions, generalizations

Keywords:

asymptotic quasi-inverse function; regularly varying function

Citations:

Zbl 1103.26001; Zbl 1101.26003