On Lower Tail Probabilities of Positive Random Sums |
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Authors: | Email author" target="_blank">J?M?P?AlbinEmail author |
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Institution: | (1) Center for Stochastic Processes, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA;(2) Department of Mathematics, Chalmers University of Technology, 412 96 Göteborg, Sweden |
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Abstract: | Let
be a sequence of independent identically distributed positive random variables with O-regularly varying distribution F at 0. Given a sequence
of positive numbers, we show that
belongs to the Type I domain of attraction of extremes for minima, by means of relating the asymptotic behaviour of P{S < } as 0, to that of E{e-S/}. Our contribution is that we dispense with the unnatural moment condition from the literature, that F has finite variance. This in turn permits a novel application to lower tails of -stable distributions on Hilbert space.AMS 2000 Subject Classification. Primary—60G50, 60G70, Secondary—60B12, 60E07, 60F05, 60G52Research supported by NFR Grant M 650-19981841/2000, and by M.R. Leadbetter |
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Keywords: | domain of attraction Esscher transform random sum regular variation stable distribution |
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