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.

我们研究了在很小的时间内出现极大值的极端过程的行为，从而获得了Fisher-Tippet-Gnedenko定理的类似物。因此，建立了将线性最大值归一化为Fréchet或Gumbel分布的此类最大值的局部收敛的充要条件。在这种情况下，威布尔分布不是可能的限制。此外，假设二阶规则变化，我们证明了中间阶统计量的局部渐近正态性，并为回火稳定过程的归一化常数导出了明确的公式。我们将希尔的尾部索引的估计器调整为较小的时间设置，并在二阶规则变化条件下建立其渐近正态性，并通过仿真进行说明。适用于资产退货流程的精细结构，