Another characterization of the type I extreme value distribution
Let (1) \(Y_ n=X_ n+cn\), \(n=1,2,...\), \(c>0\), where \(\{X_ n\}\) is an i.i.d. sequence. In the special case when \(\{X_ n\}\) is i.i.d. with type I extreme value (or Gumbel) distribution \[ \Lambda_{\alpha,\beta}(x)=\exp \{-\exp \{-\beta^{-1}(x- \alpha)\}\},\quad -\infty <x<\infty, \] the author and \textit{S. I. Resnick} [Embedding sequences of successive maxima in extremal processes. J. Appl. Probab. (to appear)] showed that the maximum \(M_ n=\max \{Y_ 1,Y_ 2,...,Y_ n\}\) is independent of the indicator of a record at time n \(1_ n=1_{[Y_ n>M_{n-1}]}\) for each n. In this paper, the author proves that independence of \(M_ n\) and \(1_ n\) for models of the form (1) holds only when \(X_ i's\) have a Gumbel distribution.
scientific article; zbMATH DE number 3998964