Limit Laws for Norms of IID Samples with Weibull Tails

Leonid Bogachev ()
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Leonid Bogachev: University of Leeds

Journal of Theoretical Probability, 2006, vol. 19, issue 4, 849-873

Abstract: We are concerned with the limit distribution of l t -norms $${R_{N}(t)=\|\mathbf X_N\|_t}$$ (of order t) of samples $${\mathbf X_{N}=(X_1,\dots,X_N)}$$ of i.i.d. positive random variables, as N→∞, t→∞. The problem was first considered by Schlather [(2001), Ann. Probab. 29, 862–881], but the case where {X i } belong to the domain of attraction of Gumbel’s double exponential law (in the sense of extreme value theory) has largely remained open (even for an exponential distribution). In this paper, it is assumed that the log-tail distribution function $${h(x)=-{\rm log P}\{X_1\ > x\}}$$ is regularly varying at infinity with index $${{0

Keywords: Sums of independent random variables; weak limit theorems; central limit theorem; infinitely divisible laws; stable laws; l-norms; Primary: 60G50; Primary: 60F05; Secondary: 60E07 (search for similar items in EconPapers)
Date: 2006
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DOI: 10.1007/s10959-006-0036-z

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