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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

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


Authors: V. V. Buldygin, O. I. Klesov and J. G. Steinebach
Translated by: The authors
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 70 (2004).
Journal: Theor. Probability and Math. Statist. 70 (2005), 11-28
MSC (2000): Primary 26A12; Secondary 60F15
Published electronically: August 5, 2005
MathSciNet review: 2109819
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Abstract: We introduce the notions of asymptotic quasi-inverse functions and asymptotic 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 asymptotic 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.


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Additional Information

V. V. Buldygin
Affiliation: Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (KPI), Peremogy 37, Kyiv 03056, Ukraine
Email: valbuld@comsys.ntu-kpi.kiev.ua

O. I. Klesov
Affiliation: Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (KPI), Peremogy 37, Kyiv 03056, Ukraine
Email: oleg@tbimc.freenet.kiev.ua

J. G. Steinebach
Affiliation: Universität zu Köln, Mathematisches Institut, Weyertal 86–90, D–50931 Köln, Germany
Email: jost@math.uni-koeln.de

DOI: http://dx.doi.org/10.1090/S0094-9000-05-00627-7
PII: S 0094-9000(05)00627-7
Received by editor(s): September 14, 2003
Published electronically: August 5, 2005
Additional Notes: This work was partially supported by Deutsche Forschungsgemeinschaft under DFG grants 436 UKR 113/41/0-2 and 436 UKR 113/68/0-1.
Article copyright: © Copyright 2005 American Mathematical Society