On the locations of maxima and minima in a sequence of exchangeable random variables

Author:
D. Ferger

Journal:
Theor. Probability and Math. Statist. **105** (2021), 35-50

MSC (2020):
Primary 60G70; Secondary 62E15

DOI:
https://doi.org/10.1090/tpms/1154

Published electronically:
December 7, 2021

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Abstract: We show for a finite sequence of exchangeable random variables that the locations of the maximum and minimum are independent from every symmetric event. In particular they are uniformly distributed on the grid without the diagonal. Moreover, for an infinite sequence we show that the extrema and their locations are asymptotically independent. Here, in contrast to the classical approach we do not use affine-linear transformations. Moreover it is shown how the new transformations can be used in extreme value statistics.

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*Limit laws for the maximum and minimum of stationary sequences*, Z. Wahrscheinlichkeitstheorie verw. Gebiete **61** (1982), 31–42. MR **671241**
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*Limit laws for upper and lower extremes from stationary mixing sequences*, J. Multivar. Anal. **13** (1983), 273–286. MR **705551**
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*On upper and lower extremes in stationary sequences*, Statistical Extremes and Applications (Vimeiro Conference), 1984, pp. 443–460. MR **784836**
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*Extreme Value Theory, an Introduction*, Springer-Verlag, New York, 2006. MR **2234156**
- R. A. Fisher and L. H. C. Tippett,
*Limiting forms of the frequency distribution of the largest or smallest member of a sample*, Proc. Camb. Phil. Soc. **24** (1928), 180–190.
- M. Fréchet,
*Sur la loi de probabilité de l’écart maximum*, Ann. Soc. Math. Polon **6** (1927), 93–116.
- J. Galambos,
*The Asymptotic Theory of Extreme Order Statistics*, Robert E. Krieger Publishing Company, Malabar, Florida, 1987. MR **936631**
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*Sur la distribution limite du terme maximum d’une série aléatoire*, Ann. Math. **44** (1943), 423–453. MR **8655**
- R. Habibi,
*Exact distribution of argmax (argmin)*, Economic Quality Control **26** (2011), 155–162. MR **2868657**
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*The rate of convergence of normal extremes*, J. Appl. Prob. **16** (1979), 433–439. MR **531778**
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*Foundations of Modern Probability*, Springer-Verlag, New York, Berlin, Heidelberg, 1997. MR **1464694**
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*Extremes and Related Properties of Random Sequences and Processes*, Springer-Verlag, Berlin, 1983. MR **691492**
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*Theory of Point Estimation*, Springer-Verlag, New York, 1983. MR **1451376**
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*Strong and weak representations of cumulative hazard function and Kaplan–Meier estimators on increasing sets*, J. Statist. Plann. Inference **42** (1994), 315–329. MR **1309627**
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Additional Information

**D. Ferger**

Affiliation:
Fakultät Mathematik, Technische Universität Dresden, D-01069 Dresden, Germany

Email:
dietmar.ferger@tu-dresden.de

Keywords:
Extreme value theory,
exchangeability,
conditional independence

Received by editor(s):
May 1, 2021

Published electronically:
December 7, 2021

Article copyright:
© Copyright 2021
Taras Shevchenko National University of Kyiv