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Some properties of rapidly varying sequences. (English) Zbl 1116.26002

Following the definition of the class of rapidly varying functions, denoted by \({\mathbb R}_{\infty ,f}\) [N. H. Bingham, C. M. Goldie and J. L. Teugels, “Regular variation” (1987; Zbl 0617.26001), § 2.4], the authors define the class of rapidly varying sequences, denoted by \({\mathbb R}_{\infty ,s}\) as follows: A sequence \((c_{n})\) of positive numbers is said to be rapidly varying if
\[ \lim _{n\rightarrow \infty } \frac{c_{[\lambda n]}}{c_{n}} = 0 \quad \text{for \; every} \quad 0<\lambda < 1 . \]
It is proved that the following two statements are equivalent:
(a) \((c_{n})\in {\mathbb R}_{\infty ,s}\),
(b) for \(x\geq 1\), \(f(x) = c_{[x]}\in {\mathbb R}_{\infty ,f}\).
A sketchy proof of a simpler (as formulated) version of this result is given earlier by the first and the third author [Math. Morav. 8, No. 2, 1-3 (2004; Zbl 1105.26001)]. In addition, the authors prove several properties of rapidly varying sequences, in particular the ones relating these to some selection principles with consequences in game theory and to the Ramsey theory. This reflects some previous investigations of the second author [e.g., “Selected results on selection principles”. Proc. 3rd Seminar on Geometry and Topology, Tabriz, Iran, 2004, 71–104).

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
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References:

[1] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0617.26001
[2] Bojanić, R.; Seneta, E., A unified theory of regularly varying sequences, Math. Z., 134, 91-106 (1973) · Zbl 0256.40002
[3] Djurčić, D.; Božin, V., A proof of a S. Aljančić hypothesis on \(O\)-regularly varying sequences, Publ. Inst. Math. (Beograd), 62, 76, 46-52 (1997) · Zbl 0946.26002
[4] Djurčić, D.; Torgašev, A., Representation theorem for the sequences of the classes \(C R_c\) and \(E R_c\), Siberian Math. J., 45, 5, 834-838 (2004)
[5] Djurčić, D.; Torgašev, A., On the Seneta sequences, Acta Math. Sin. (Engl. Ser.), 22, 3 (2006) · Zbl 1170.26300
[6] Galambos, J.; Seneta, E., Regularly varying sequences, Proc. Amer. Math. Soc., 41, 110-116 (1973) · Zbl 0247.26002
[7] de Haan, L., On Regular Variations and Its Applications to the Weak Convergence of Sample Extremes, CWI Tract, vol. 32 (1970), Math. Centrum: Math. Centrum Amsterdam
[8] Kočinac, Lj. D.R., Generalized Ramsey theory and topological properties: A survey, Proceedings of the International Symposium on Graphs, Designs and Applications. Proceedings of the International Symposium on Graphs, Designs and Applications, Messina, September 30-October 4, 2003. Proceedings of the International Symposium on Graphs, Designs and Applications. Proceedings of the International Symposium on Graphs, Designs and Applications, Messina, September 30-October 4, 2003, Rend. Sem. Mat. Messina Ser. II, 25, 9, 119-132 (2003) · Zbl 1124.05089
[9] Lj.D.R. Kočinac, Selected results on selection principles, in: Sh. Rezapour (Ed.), Proceedings of the 3rd Seminar on Geometry and Topology, July 15-17, 2004, Tabriz, Iran, pp. 71-104; Lj.D.R. Kočinac, Selected results on selection principles, in: Sh. Rezapour (Ed.), Proceedings of the 3rd Seminar on Geometry and Topology, July 15-17, 2004, Tabriz, Iran, pp. 71-104
[10] Lj.D.R. Kočinac, Selection principles related to \(\operatorname{Α;}_i\) properties, preprint; Lj.D.R. Kočinac, Selection principles related to \(\operatorname{Α;}_i\) properties, preprint
[11] Kočinac, Lj. D.R.; Scheepers, M., Combinatorics of open covers (VII): Groupability, Fund. Math., 179, 2, 131-155 (2003) · Zbl 1115.91013
[12] Tasković, M., Fundamental facts on translational \(O\)-regularly varying functions, Math. Moravica, 7, 107-152 (2003) · Zbl 1274.26005
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