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On lower tail proba...
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Albin, Patrik,1960Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper,Department of Mathematical Sciences
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On lower tail probabilities of positive random sums
- Article/chapterEnglish2005
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LIBRIS-ID:oai:gup.ub.gu.se/98079
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https://gup.ub.gu.se/publication/98079URI
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Let $(\xi_k)_{k\geq1}$ be an i.i.d. sequence of positive random variables with $O$-varying distribution function at 0. Further let $(a_k)_{k\geq1}$ be a sequence of positive weights such that the positive random sum $S=\sum_{k=1}^\infty a_k\xi_k$ exists almost surely. Without assuming finite second moments, the author determines the asymptotic behaviour of the left tail $P\{S<\varepsilon\}$ as $\varepsilon\downarrow0$ and of the density function at 0 in terms of the asymptotic behaviour of the Laplace transform at $\infty$ using Escher transforms. It turns out that necessarily $S$ belongs to the Type I domain of attraction for minima. An application is given for $\xi_k=\eta_k^2$ with $\alpha$-stable random variables $\eta_k$ on the real line.
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Göteborgs universitetInstitutionen för matematiska vetenskaper
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In:Extremes7:3, s. 199-2201386-1999
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