The possibility of existence of extremal indices exceeding one

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).