Power variation for a class of stationary increments Lévy driven moving averages

Abstract

In this paper, we present some new limit theorems for power variation of kth order increments of stationary increments Lévy 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 BlumenthalGetoor index β ∈ [0, 2) of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0 determined by the power α. 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 Lévy process L is a symmetric β-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k − α)β-stable totally right skewed random variable.

On critical cases in limit theory for stationary increments Lévy driven moving averages

“…We refer e.g. to [4,12,13,15] for limit theory for power variations of Itô semimartingales, to [2,3,9,11,14] for the asymptotic results in the framework In a recent paper [5] the power variation of stationary increments Lévy driven moving averages has been studied. Let us recall the definitions, notations and main results of this paper.…”

“…In order to describe the main results of [5] we need to introduce some notation and a set of assumptions. First of all, we consider the kth order increments ∆ n i,k X of X, k ∈ N, that are defined by where g(t) ∼ f (t) as t ↓ 0 means that lim t↓0 g(t)/f (t) = 1.…”

“…The following first order limit theory for the power variation V (p; k) n has been proved in [5]. We refer to [1,17] for the definition of F-stable convergence in law which will be denoted L−s −→.…”

“…Moreover, P −→ will denote convergence in probability. Theorem 1.1 (First order asymptotics [5]). Suppose (A) is satisfied and assume that the Blumenthal-Getoor index satisfies β < 2.…”

“…The limit theory heavily depends on the interplay between the given order of the increments, the considered power, the Blumenthal-Getoor index of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0. In this work we will study the critical cases, which were not covered in the original work [5]. …”