Abstract
The classical extremal index is an important characteristic of the asymptotic behavior of maxima in stationary random sequences. However, in practice it is also necessary to study maxima on more complex structures than the set of natural numbers. This paper continues the cycle of work devoted to the author’s generalization of the extremal index to a scheme of random-length series. This generalization allows working with a wider class of stochastic structures. For cases where there is no exact extremal index, partial indices were introduced earlier. Unlike the classical extremal index, they can take values greater than one (which corresponds to a negative dependence of random variables). The question whether the exact extremal index greater than one can exist remains open. In the paper, this question is partially closed (the impossibility is proved under certain conditions).
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Notes
In what follows, we denote by \(\vee\) the maximum and by \(\wedge\) the minimum, and a bar over a distribution function denotes its tail: \({\bar{F}}(x)=1-F(x)\).
As compared to Definition A, the change of variables \(s=e^{-\tau}\) is made and accordingly the functions \(u_{n}\) (\(n\geqslant 1\)) are redefined.
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Translated by I. Tselishcheva
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Lebedev, A.V. The Possibility of Existence of Extremal Indices Exceeding One. Moscow Univ. Math. Bull. 77, 1–6 (2022). https://doi.org/10.3103/S0027132222010041
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DOI: https://doi.org/10.3103/S0027132222010041