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Principle Seminar of the Department of Probability Theory, Moscow State University
April 19, 2017 16:45–17:05, Moscow, MSU, auditorium 12-24

On extremal indices which are greater than one for maxima in the series scheme

A. A. Goldaeva, A. V. Lebedev

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract: The classical extremal index $\theta$ is an important characteristic of the asymptotic behavior of maxima of stationary random sequences. One of the interpretations of the extremal index is that the excess of a high level in the sequence occurs not singly, but by groups (clusters) with the average size $1/\theta$. In applications, this can mean natural disasters, failures of technical systems, loss of data in the transmission of information, financial losses, etc. It is clear that when such events occur several times in a row, it is much more dangerous than single cases and should be taken into account in risk management. However, in practice, there is also a need to study maxima on more complex structures than the set of natural numbers. For example, if we are talking about the lifetimes of the components of a complex system (reliability scheme), then it is not clear how to number them so as to use the model of a stationary sequence. A.V.Lebedev (2015) gives generalizations of the extremal index to the series scheme with random lengths, which allows us to work with a wider class of stochastic structures. For cases when there is no exact extremal index, partial indices (upper and lower, left and right) are introduced. It turns out that unlike the classical extremal index, they can take values greater than one (which corresponds to the negative dependence of random variables). In this report, a new model is considered in which the left and right indices take on values greater than one, including ones that can be equal, but there is no exact index.