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 Levy 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 \(n^{1-\frac{p}{\alpha}}.\)