We study the conditions of convergence to infinity for some classes of functions extending the well-known class of regularly varying (RV) functions, such as, e.g., O-regularly varying (ORV) functions or positive increasing (PI) functions.
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S. Aljančić and D. Arandelović, “O-regularly varying functions,” Publ. Inst. Math. (Beograd) (N.S.), 22(36), 5–22 (1977).
D. Arandelović, “O-regular variation and uniform convergence,” Publ. Inst. Math. (Beograd) (N.S.), 48(62), 25–40 (1990).
V. G. Avakumović, “Über einen O-Inversionssatz,” Bull. Int. Acad. Youg. Sci., 29–30, 107–117 (1936).
N. K. Bari and S. B. Stechkin, “Best approximation and differential properties of two conjugate functions,” Tr. Mosk. Mat. Obsch., 5, 483–522 (1956).
N. H. Bingham and C. M. Goldie, “Extensions of regular variation, I, II,” Proc. London Math. Soc., 44, 473–534 (1982).
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge (1987).
V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, “Properties of a subclass of Avakumović functions and their generalized inverses,” Ukr. Math. J., 54, No. 2, 179–206 (2002).
V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, “Some properties of asymptotically quasiinverse functions and their applications. I,” Theory Probab. Math. Statist., 70, 11–28 (2005).
V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, “On some properties of asymptotically quasiinverse functions and their applications. II,” Theory Probab. Math. Statist., 71, 37–52 (2005).
V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, “PRV property and the asymptotic behavior of solutions of stochastic differential equations,” Theory Stochast. Proc., 11(27), 42–57 (2005).
V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, “On some properties of asymptotically quasiinverse functions,” Theory Probab. Math. Statist., 77, 15–30 (2008).
V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, “PRV property and the φ-asymptotic behavior of solutions of stochastic differential equations,” Liet. Mat. Rink., 47, No. 4, 445–465 (2007).
D. Djurčić and A. Torgašev, “Strong asymptotic equivalence and inversion of functions in the class K,” J. Math. Anal. Appl., 255, 383–390 (2001).
W. Feller, “One-sided analogues of Karamata’s regular variation,” L’Enseignement Math., 15, 107–121 (1969).
I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer, Berlin (1972).
W. Greub, Linear Algebra, 4th ed., Springer, Berlin (1975).
L. de Haan and U. Stadtmüller, “Dominated variation and related concepts and Tauberian theorems for Laplace transformations,” J. Math. Anal. Appl., 108, 344–365 (1985).
J. Karamata, “Sur un mode de croissance régulière des fonctions,” Mathematica (Cluj), 4, 38–53 (1930).
J. Karamata, “Sur un mode de croissance régulière. Théoremès fondamentaux,” Bull. Soc. Math. France, 61, 55–62 (1933).
J. Karamata, “Bemerkung über die vorstehende Arbeit des Herrn Avakumović, mit näherer Betrachtung einer Klasse von Funktionen, welche bei den Inversionssätzen vorkommen,” Bull. Int. Acad. Youg. Sci., 29–30, 117–123 (1936).
G. Keller, G. Kersting, and U. Rösler, “On the asymptotic behavior of solutions of stochastic differential equations,” Z. Wahrscheinlichkeitstheor. verw. Geb., 68, 163–184 (1984).
J. Korevaar, T. van Aardenne-Ehrenfest, and N. G. de Bruijn, “A note on slowly oscillating functions,” Nieuw Arch. Wisk., 23, 77–86 (1949).
B. A. Rogozin, “A Tauberian theorem for increasing functions of dominated variation,” Sib. Math. J., 43, 353–356 (2002).
E. Seneta, Regularly Varying Functions, Springer, Berlin (1976).
A. L. Yakymiv, “Asymptotics properties of the state change points in a random record process,” Theory Probab. Appl., 31, 508–512 (1987).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 10, pp. 1299–1308, October, 2010.
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Buldygin, V.V., Klesov, O.I. & Steinebach, J.G. On the convergence of positive increasing functions to infinity. Ukr Math J 62, 1507–1518 (2011). https://doi.org/10.1007/s11253-011-0446-7
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DOI: https://doi.org/10.1007/s11253-011-0446-7