Convergence of extreme values of Poisson point processes at small times

We study the behaviour of large values of extremal processes at small times, obtaining an analogue of the Fisher-Tippet-Gnedenko Theorem. Thus, necessary and sufficient conditions for local convergence of such maxima, linearly normalised, to the Fréchet or Gumbel distributions, are established. Weibull distributions are not possible limits in this situation. Moreover, assuming second order regular variation, we prove local asymptotic normality for intermediate order statistics, and derive explicit formulae for the normalising constants for tempered stable processes. We adapt Hill’s estimator of the tail index to the small time setting and establish its asymptotic normality under second order regular variation conditions, illustrating this with simulations. Applications to the fine structure of asset returns processes, possibly with infinite variation, are indicated.