More on Translationally Slowly Varying Sequences

Abstract

We define and study an equivalence relation in the class ${\sf Tr(SV_s)}$ of translationally slowly varying positive real sequences and its relations with selection principles and game theory. We also prove a game-theoretic result for translationally rapidly varying sequences.

Keywords

• Rapid equivalence,Class $R_{s;\infty}$,Selection principles

References

1. [1] N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, CambridgeUniv. Press, Cambridge, 1987.
2. [2] D. Djurcic, Lj.D.R. Kocinac, M.R. Zizovic, Some properties of rapidly varying sequences, J. Math. Anal. Appl. 327:2 (2007) 1297-1306.
3. [3] D. Djuric, Lj.D.R. Kocinac, M.R. Zizovic, Classes of sequences of real numbers, games and selection properties, Topology Appl. 156:1 (2008) 46-55.
4. [4] D. Djurcic, Lj.D.R. Kocinac, M.R. Zizovic, A few remarks on divergent sequences: Rates of divergence, J. Math. Anal. Appl. 360:2 (2009) 588-598.
5. [5] D. Djurcic, Lj.D.R. Kocinac, M.R. Zizovic, A few remarks on divergent sequences: rates of divergence II, J. Math. Anal. Appl. 367:2 (2010) 705-709.
6. [6] D. Djurcic, A. Torgasev, S. Jesic, The strong asymptotic equivalence and the generalized inverse, Siber. Math. J. 49:4 (2008) 786-795.
7. [7] L. de Haan, On Regular Variation and its Applications to the Weak Convergence of Sample Extremes, Math. Centre Tracts, Vol. 32, CWI, Amsterdam, 1970.
8. [8] J. Karamata, Sur un mode de croissance reguliere des functions, Mathematica(Cluj) 4 (1930) 38-53.

9. [9] Lj.D.R. Kocinac, Selected results on selection principle, In: Proc. Third.Sem. Geom. Topology, Tabriz, Iran (2004) 71-104.
10. [10] Lj.D.R. Kocinac, On the ®i-selection principles and games, Cont. Math.533 (2011) 107-124.
11. [11] Lj.D.R. Kocinac, D. Djurcic, J.V. Manojlovic, Regular and Rapid Variations and Some Applications, In: M. Ruzhansky, H. Dutta, R.P. Agarwal(eds.), Mathematical Analysis and Applications: Selected Topics, Chapter12, John Wiley & Sons, Inc., (2018) 414-474.
12. [12] V. Timotic, D. Djurcic, R.M. Nikolic, On slowly varying sequences, Filomat29:1 (2015) 7-12