Geodesic
The possibility of existence of extremal indices exceeding one
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2022), pp. 3-8 Cet article a éte moissonné depuis la source Math-Net.Ru

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The classical extremal index is an important characteristic of the asymptotic behavior of maxima in stationary random sequences. However, in practice, there is also a need to study maxima on more complex structures than the natural numbers set. This paper continues the cycle devoted to the author's generalization of the extremal index to a random-length series scheme, which allows working with a wider class of stochastic structures. For cases where an exact extremal index does not exist, partial indices were previously introduced. Unlike the classical extremal index, they can take values greater than one (which corresponds to a negative dependence of random variables). The question is whether an exact extremal index greater than one is possible remains open. In this paper, this question is partially closed (the impossibility is proved under certain conditions). 
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A. V. Lebedev. The possibility of existence of extremal indices exceeding one. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2022), pp. 3-8. http://geodesic.mathdoc.fr/item/VMUMM_2022_1_a0/

[1] Lidbetter M., Lindgren G., Rotsen Kh., Ekstremumy sluchainykh posledovatelnostei i protsessov, Mir, M., 1989 | MR

[2] Embrechts P., Klüppelberg C., Mikosh T., Modelling extremal events for insurance and finance, Springer, Berln, 2003 | MR

[3] de Haan L., Ferreira A., Extreme value theory. An introduction, Springer, N.Y., 2006 | MR | Zbl

[4] Novak S.Yu., Extreme value methods with applications to finance, Monographs on Statistics and Applied Probability, 122, CRC Press, Boca Raton, FL, 2012 | MR | Zbl

[5] Lebedev A. V., “Ekstremalnye indeksy v skheme serii i ikh prilozheniya”, Inform. i ee primen., 9:3 (2015), 39–54

[6] Lebedev A. V., Neklassicheskie zadachi stokhasticheskoi teorii ekstremumov, Dokt. dis., MGU, M., 2016

[7] Goldaeva A. A., Lebedev A. V., “On extremal indices greater than one for a scheme of series”, Lithuan. Math. J., 58:4 (2018), 384–398 | DOI | MR | Zbl

[8] Markovich N. M., Rodionov I. V., “Maxima and sums of non-stationary random length sequences”, Extremes, 23:3 (2020), 451–464 | DOI | MR | Zbl

[9] Markovich N. M., Extremes of sums and maxima with application to random networks, arXiv: 2110.04120

[10] Nelsen R., An introduction to copulas, Springer, N.Y., 2006 | MR | Zbl

[11] McNeil A. J., Frey R., Embrechts P., Quantitative risk management, Princeton University Press, Princeton, 2005 | MR | Zbl

[12] Blagoveschenskii Yu. N., “Osnovnye elementy teorii kopul”, Prikl. ekonometrika, 26:2 (2012), 113–130 | MR