Stable CLTs and Rates for Power Variation of α-Stable Lévy Processes

Abstract

In a central limit type result it has been shown that the pth power variations of an α-stable Lévy process along sequences of equidistant partitions of a given time interval have $\frac{\alpha}{p}$-stable limits. In this paper we give precise orders of convergence for the distances of the approximate power variations computed for partitions with mesh of order $\frac{1}{n}$ and the limiting law, measured in terms of the Kolmogorov-Smirnov metric. In case 2α < p the convergence rate is seen to be of order $\frac{1}{n}$, in case α < p < 2α the order is $n^{1-\frac{p}{\alpha}}.$