Tails of waiting times and their bounds |
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Authors: | Kalashnikov Vladimir Tsitsiashvili Gurami |
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Institution: | (1) Institute for Information Transmission Problems, Bol’shoi, Karetny 19, 101447 Moscow, Russia;(2) Institute of Applied Mathematics, Radio st. 7, 690041 Vladivostok 41, Russia |
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Abstract: | Tails of distributions having the form of the geometric convolution are considered. In the case of light-tailed summands,
a simple proof of the famous Cramér asymptotic formula is given via the change of probability measure. Some related results
are obtained, namely, bounds of the tails of geometric convolutions, expressions for the distribution of the 1st failure time
and failure rate in regenerative systems, and others. In the case of heavy-tailed summands, two-sided bounds of the tail of
the geometric convolution are given in the cases where the summands have either Pareto or Weibull distributions. The results
obtained have the property that the corresponding lower and upper bounds are tailed-equivalent.
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | the Cramér condition change of probability measure asymptotic formula renewal process two-sided bounds |
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