## 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 $$n\geq 1$$.

### MSC:

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

subexponential
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### References:

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