Electronic Journal of Probability

On limit theory for functionals of stationary increments Lévy driven moving averages

Andreas Basse-O’Connor, Claudio Heinrich, and Mark Podolskij

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Abstract

In this paper we present new limit theorems for variational functionals of stationary increments Lévy driven moving averages in the high frequency setting. More specifically, we will show the “law of large numbers” and a “central limit theorem”, which heavily rely on the kernel, the driving Lévy process and the properties of the functional under consideration. The first order limit theory consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we prove the associated weak limit theory, which again consists of three different cases. Our work is related to [10, 7], who considered power variation functionals of stationary increments Lévy driven moving averages. However, the asymptotic theory of the present paper is more complex. In particular, the weak limit theorems are derived for an arbitrary Appell rank of the involved functional.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 79, 42 pp.

Dates
Received: 23 July 2018
Accepted: 18 June 2019
First available in Project Euclid: 5 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1567648850

Digital Object Identifier
doi:10.1214/19-EJP336

Subjects
Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60G22: Fractional processes, including fractional Brownian motion 60G52: Stable processes

Keywords
fractional processes limit theorems self-similarity stable processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Basse-O’Connor, Andreas; Heinrich, Claudio; Podolskij, Mark. On limit theory for functionals of stationary increments Lévy driven moving averages. Electron. J. Probab. 24 (2019), paper no. 79, 42 pp. doi:10.1214/19-EJP336. https://projecteuclid.org/euclid.ejp/1567648850


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