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Limit theorems for counting variables based on records and extremes
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Univ Ulm, Dept Number Theory & Probabil Theory, D-89069 Ulm, Germany..
2017 (English)In: Extremes, ISSN 1386-1999, E-ISSN 1572-915X, Vol. 20, no 1, p. 33-52Article in journal (Refereed) Published
Abstract [en]

Hsu and Robbins (Proc. Nat. Acad. Sci. USA 33, 25-31, 1947) introduced the concept of complete convergence as a complement to the Kolmogorov strong law, in that they proved that provided the mean of the summands is zero and that the variance is finite. Later, ErdoIis proved the necessity. Heyde (J. Appl. Probab. 12, 173-175, 1975) proved that, under the same conditions, , thereby opening an area of research which has been called precise asymptotics. Both results above have been extended and generalized in various directions. Some time ago, Kao proved a pointwise version of Heyde's result, viz., for the counting process , he showed that , where W(ai...) is the standard Wiener process. In this paper we prove analogs for extremes and records for i.i.d. random variables with a continuous distribution function.

Place, publisher, year, edition, pages
SPRINGER , 2017. Vol. 20, no 1, p. 33-52
Keywords [en]
Record times, Records, Extremes, Counting process, Weak convergence
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-319544DOI: 10.1007/s10687-016-0269-xISI: 000394354400002OAI: oai:DiVA.org:uu-319544DiVA, id: diva2:1087227
Available from: 2017-04-06 Created: 2017-04-06 Last updated: 2017-11-29Bibliographically approved

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