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Asymptotics of joint maxima for discontinuous random variables. (English) Zbl 1226.60080

Summary: This paper explores the joint extreme-value behavior of discontinuous random variables. It is shown that as in the continuous case, the latter is characterized by the weak limit of the normalized componentwise maxima and the convergence of any compatible copula. Illustrations are provided and an extension to the case of triangular arrays is considered which sheds new light on recent work of S. Coles and F. Pauli [Stat. Probab. Lett. 54, No. 4, 373–379 (2001; Zbl 1026.60068)] and K. Mitov and S. Nadarajah [Extremes 8, No. 4, 357–370 (2005; Zbl 1142.60354)]. This leads to considerations on the meaning of the bivariate upper tail dependence coefficient of H. Joe [Comput. Stat. Data Anal. 16, No. 3, 279–297 (1993; Zbl 0937.62601)] in the discontinuous case.

MSC:

60G70 Extreme value theory; extremal stochastic processes
62E20 Asymptotic distribution theory in statistics
62H05 Characterization and structure theory for multivariate probability distributions; copulas

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References:

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