Abstract
Increments of the renewal function related to the distributions with infinite means and regularly varying tails of orders α ∈ (0,1] were described by Erickson in 1979 (Trans. Amer. Math. Soc. 151: 263–291, 1970). However, explicit asymptotics for the increments are known for α ∈ (½,1] only. For smaller α one can get, generally speaking, only the lower limit of the increments. There are many examples showing that this statement cannot be improved in general.
We refine Erikson's results by describing sufficient conditions for regularity of the renewal measure density of the distributions with regularly varying tails with α ∈ (0,½]. We also discuss the reasons of non-regular behavior of the renewal function increments in the general situation.
Mathematics Subject Classification (2000): 60K05, 60E07, 60J80
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References
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Acknowledgements
The author thanks Prof. Vladimir Vatutin and Prof. Dmitrii Korshunov for their comments about the initial version of this article. Their suggestions allow me to improve considerably the presentation of results. This work was supported by the Foundation of the President of the Russian Federation (Grant 3695.2008.1).
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Topchii, V. (2010). Renewal measure density for distributions with regularly varying tails of order a α ∈ (0,½]. In: González Velasco, M., Puerto, I., Martínez, R., Molina, M., Mota, M., Ramos, A. (eds) Workshop on Branching Processes and Their Applications. Lecture Notes in Statistics(), vol 197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11156-3_8
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DOI: https://doi.org/10.1007/978-3-642-11156-3_8
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