Arbelaez, H., Hernandez, R., & Sierra, W. (2022). Lower and upper order of harmonic mappings. J. Math. Anal. Appl., 507(2), 125837.
Abstract: In this paper, we define both the upper and lower order of a sense-preserving harmonic mapping in D. We generalize to the harmonic case some known results about holomorphic functions with positive lower order and we show some consequences of a function having finite upper order. In addition, we improve a related result in the case when there is equality in a known distortion theorem for harmonic mappings with finite upper order. Some examples are provided to illustrate the developed theory. (C) 2021 Elsevier Inc. All rights reserved.
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Barrera, J., & Lagos, G. (2020). Limit distributions of the upper order statistics for the Levy-frailty Marshall-Olkin distribution. Extremes, 23, 603–628.
Abstract: The Marshall-Olkin (MO) distribution is considered a key model in reliability theory and in risk analysis, where it is used to model the lifetimes of dependent components or entities of a system and dependency is induced by “shocks” that hit one or more components at a time. Of particular interest is the Levy-frailty subfamily of the Marshall-Olkin (LFMO) distribution, since it has few parameters and because the nontrivial dependency structure is driven by an underlying Levy subordinator process. The main contribution of this work is that we derive the precise asymptotic behavior of the upper order statistics of the LFMO distribution. More specifically, we consider a sequence ofnunivariate random variables jointly distributed as a multivariate LFMO distribution and analyze the order statistics of the sequence asngrows. Our main result states that if the underlying Levy subordinator is in the normal domain of attraction of a stable distribution with index of stability alpha then, after certain logarithmic centering and scaling, the upper order statistics converge in distribution to a stable distribution if alpha> 1 or a simple transformation of it if alpha <= 1. Our result can also give easily computable confidence intervals for the last failure times, provided that a proper convergence analysis is carried out first.
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