On some features of asymptotically quasi-inverse functions and their applications. I. (Ukrainian, English) Zbl 1103.26001

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

Teor. Jmovirn. Mat. Stat. 70, 9-25 (2004); translation in Theory Probab. Math. Stat. 70, 11-28 (2005).

A function \(\tilde f^{(-1)}\) is called asymptotically quasi inverse (a.q.i.) to \(f: R\to R\) if \(f(\tilde f^{(-1)}(t))\sim t\) as \(t\to\infty\) and asymptotically inverse if it is a.q.i. and \(\tilde f^{(-1)}(f(t))\sim t\). For \(c>0\) denote \(f^*(c)=\lim\sup_{t\to\infty} f(ct)/f(t)\), \(f_*(c)=\lim\inf_{t\to\infty} f(ct)/f(t)\). A function \(f\) is called pseudo regularly varying (PRV) if \(\lim\sup_{c\to 1} f^*(c)=1\) (for regularly varying functions \(f^*(c)\to 1\) as \(c\to 1\)). A PRV function \(f\) is called POV if \(f_*(c)>1\) for all \(c>1\). Different features of PRV and POV functions are investigated. E.g. it is shown that a.q.i. functions exist for PRV functions and a.i. functions exist for POV functions.

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

26A12	Rate of growth of functions, orders of infinity, slowly varying functions
60F15	Strong limit theorems

Keywords:

pseudo regularly varying function; positive order of variation; asymptotic equivalence; POV, PRV