Abstract
Considering geometrically distributed random variables thed-maximum of these events is investigated, i.e. thed-th largest element (with repetitions allowed). The quantitative behaviour of expectation and variance is analyzed thoroughly. In particular the asymptotics of the variance ford getting large is established by means of nontrivial techniques from combinatorial analysis and complex variable theory. These results apply to probabilistic counting algorithms, where the cardinalities of large sets are estimated.
Zusammenfassung
Bezüglich geometrisch verteilter zufälliger Veränderlicher wird dasd-Maximum solcher Ereignisse studiert, also dasd-größte Element, wobei Wiederholungen erlaubt sind. Das quantitative Verhalten von Erwartungswert und Varianz wird ausgiebig analysiert. Insbesondere wird das Verhalten der Varianz für großed mit Hilfe nichttrivialer Techniken aus Kombinatorik und komplexer Analysis untersucht. Diese Resultate haben Anwendungen bei probabilistischen Zählalgorithmen, die zur Schätzung der Kardinalitäten großer Mengen verwendet werden.
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Kirschenhofer, P., Prodinger, H. A result in order statistics related to probabilistic counting. Computing 51, 15–27 (1993). https://doi.org/10.1007/BF02243826
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DOI: https://doi.org/10.1007/BF02243826