The meta-elliptical distributions with given marginals. (English) Zbl 1002.62016

J. Multivariate Anal. 82, No. 1, 1-16 (2002); corrigendum ibid. 94, No. 1, 222-223 (2005).
Let \(\xi_1,\xi_2,\dots\) be independent random variables with the common distribution \(F\) such that \(F((-\infty,0]) <1\), and \({\mathbf E}\xi_1\) exists and equals \(-a<0\). Put \(S_0=0\), \(S_n=\xi_1+ \cdots+\xi_n\) and \(M_n=\max \{S_k,\;0\leq k\leq n\}\). The main result is as follows.
If the distribution of \(\xi_1 I\{\xi_1\geq 0\}\) is strongly subexponential, then \[ P(M_n\geq x)= \bigl(1+ \varepsilon_n(x) \bigr)a^{-1} \int_x^{x+na} F\bigl((u,\infty) \bigr) du, \] where \(\varepsilon_n(x)\to 0\) as \(x\to \infty\) uniformly in \(n\geq 1\).


62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
62H05 Characterization and structure theory for multivariate probability distributions; copulas


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