×

The maximum domain of attraction of multivariate extreme value distributions is small. (English) Zbl 1511.60077

Summary: Consider the set of Borel probability measures on \(\mathbb{R}^k\) and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60B10 Convergence of probability measures
60E05 Probability distributions: general theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anderson, C. W.: Extreme value theory for a class of discrete distributions with applications to some stochastic processes, J. Appl. Probability 7, (1970), 99-113. · Zbl 0192.54202
[2] Aveni, A. and Leonetti, P.: Most numbers are not normal, Math. Proc. Cambridge Philos. Soc., to appear. DOI: 10.1017/S0305004122000469 · Zbl 1526.11042
[3] Balcerzak, M. and Leonetti, P.: Convergent subseries of divergent series, Rend. Circ. Mat. Palermo (2) 71, (2022), no. 2, 879-886. · Zbl 1527.40001
[4] Ben Arous, G. and Kargin, V.: Free point processes and free extreme values, Probab. Theory Related Fields 147, (2010), no. 1-2, 161-183. · Zbl 1195.46070
[5] Ben Arous, G. and Voiculescu, D. V.: Free extreme values, Ann. Probab. 34, (2006), no. 5, 2037-2059. · Zbl 1117.46044
[6] Bercovici, H. and Pata, V.: Stable laws and domains of attraction in free probability theory, Ann. of Math. (2) 149, (1999), no. 3, 1023-1060, With an appendix by Philippe Biane. · Zbl 0945.46046
[7] Billingsley, P.: Convergence of probability measures. Second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication. · Zbl 0944.60003
[8] Davison, A. C., Padoan, S. A., and Ribatet, M.: Statistical modeling of spatial extremes, Statist. Sci. 27, (2012), no. 2, 161-186. · Zbl 1330.86021
[9] de Haan, L. and Ferreira, A.: Extreme value theory. Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006, An introduction. · Zbl 1101.62002
[10] Dudley, R. M.: Real analysis and probability. Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original. · Zbl 1023.60001
[11] Gnedenko, B.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. of Math. (2) 44, (1943), 423-453. · Zbl 0063.01643
[12] Hooghiemstra, G. and Greenwood, P. E.: The domain of attraction of the \(α\)-sun \([α\)-sum] operator for type II and type III distributions, Bernoulli 3, (1997), no. 4, 479-489. · Zbl 0899.60018
[13] Kechris, A. S.: Classical descriptive set theory. Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. · Zbl 0819.04002
[14] Leonetti, P.: Limit points of subsequences, Topology Appl. 263, (2019), 221-229. · Zbl 1430.40002
[15] Leonetti, P.: Tauberian theorems for ordinary convergence, J. Math. Anal. Appl. 519, (2023), no. 2, 126798. DOI: 10.1016/j.jmaa.2022.126798 · Zbl 07624102
[16] Marshall, A. W. and Olkin, I.: Domains of attraction of multivariate extreme value distributions, Ann. Probab. 11, (1983), no. 1, 168-177. · Zbl 0508.60022
[17] Oxtoby, J. C.: Measure and category. Second ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980, A survey of the analogies between topological and measure spaces. · Zbl 0435.28011
[18] Padoan, S. A., Ribatet, M., and Sisson, S. A.: Likelihood-based inference for max-stable processes, J. Amer. Statist. Assoc. 105, (2010), no. 489, 263-277. · Zbl 1397.62172
[19] Pickands III, J.: The continuous and differentiable domains of attraction of the extreme value distributions, Ann. Probab. 14, (1986), no. 3, 996-1004. · Zbl 0593.60035
[20] Resnick, S. I.: Extreme values, regular variation and point processes. Springer Series in Operations Research and Financial Engineering, Springer, New York, 2008, Reprint of the 1987 original. · Zbl 1136.60004
[21] Shimura, T.: Discretization of distributions in the maximum domain of attraction, Extremes 15, (2012), no. 3, 299-317. · Zbl 1329.60159
[22] Sweeting, T. J.: On domains of uniform local attraction in extreme value theory, Ann. Probab. 13, (1985), no. 1, 196-205. · Zbl 0566.60022
[23] Tawn, J. A.: Bivariate extreme value theory: models and estimation, Biometrika 75, (1988), no. 3, 397-415. · Zbl 0653.62045
[24] Yun, S.: On domains of attraction of multivariate extreme value distributions under absolute continuity, J. Multivariate Anal. 63, (1997), no. 2, 277-295. · Zbl 0897.60053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.