Criteria for Convergence of the Number of Near Maxima for Long Tails |
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| Authors: | Email author" target="_blank">Anthony?G?PakesEmail author |
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| Institution: | (1) School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, W. A., 6009, Australia |
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| Abstract: | Let Mn denote the maximum of a random sample of size n and Kn(a) be the number of near maxima, i.e. the number of sample observations in the fixed-width window (Mn – a, Mn]. There is a known integral criterion for almost sure convergence (to unity) of Kn(a), and we establish a similar criterion for complete convergence. We obtain simple but quite general sufficient conditions on the survivor function for satisfying the integral criteria. Further insight is obtained by seeking the rate at which P(Kn(a > 1)) tends to zero.AMS 2000 Subject Classification. 62G30, 60F15 |
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| Keywords: | almost sure and complete convergence extreme value laws local limit theorems order statistics rate of convergence regular variation |
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