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Sankhya A

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The Min-characteristic Function: Characterizing Distributions by Their Min-linear Projections

  • Michael Falk
  • Gilles StupflerEmail author
Article
  • 17 Downloads

Abstract

Motivated by a (seemingly previously unnoticed) result stating that d −dimensional distributions on \((0,\infty )^{d}\) are characterized by the collection of their min-linear projections, we introduce and study a notion of min-characteristic function (min-CF) of a random vector with strictly positive components. Unlike the related notion of max-characteristic function which has been studied recently, the existence of the min-CF does not hinge on any integrability conditions. It is itself a multivariate distribution function, which is continuous and concave, no matter which properties the initial distribution function has. We show the equivalence between convergence in distribution and pointwise convergence of min-CFs, and we also study the functional convergence of the min-CF of the empirical distribution function of a sample of independent and identically distributed random vectors. We provide some further insight into the structure of the set of min-CFs, and we conclude by showing how transforming the components of an arbitrary random vector by a suitable one-to-one transformation such as the exponential function allows the construction of a notion of min-CF for arbitrary random vectors.

Keywords and phrases

Characteristic function Copula D-norm Max-linear projections Min-linear projections Multivariate distribution. 

AMS (2000) subject classification

Primary 60E10 Secondary 62H05 

Notes

Acknowledgments

This research was in part carried out when M. Falk was visiting G. Stupfler at the University of Nottingham in July 2018. The first author is grateful to his host for his hospitality and the extremely constructive atmosphere. Support from the London Mathematical Society Research in Pairs Scheme (reference 41710) is gratefully acknowledged. The authors are indebted to an anonymous reviewer for his/her constructive remarks which led to an improved presentation of the results of the paper.

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Copyright information

© Indian Statistical Institute 2019

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany
  2. 2.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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