POWER VARIATION FOR A CLASS OF STATIONARY INCREMENTS LEVY DRIVEN MOVING AVERAGES

• 成果类型: Article
• 署名作者: Basse-O'Connor, Andreas; Lachieze-Rey, Raphael; Podolskij, Mark
• 署名单位: Aarhus University; Universite Paris Cite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1170
• 发表日期: 2017
• 页码: 4477-4528
• 关键词: infinitely divisible processes; central-limit-theorem; Empirical Processes; gaussian-processes; random-variables; stable limits; variance; SEMIMARTINGALES; functionals; continuity

摘要

In this paper, we present some new limit theorems for power variation of kth order increments of stationary increments Levy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k >= 1, the considered power p > 0, the Blumenthal-Getoor index beta is an element of [ 0, 2) of the driving pure jump Levy process L and the behaviour of the kernel function g at 0 determined by the power alpha. First-order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove a second-order limit theorem connected to the ergodic type result. When the driving Levy process L is a symmetric beta-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k - alpha) beta-stable totally right skewed random variable.