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2021 Power variations for fractional type infinitely divisible random fields
Andreas Basse-O’Connor, Vytautė Pilipauskaitė, Mark Podolskij
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Electron. J. Probab. 26: 1-35 (2021). DOI: 10.1214/21-EJP617


This paper presents new limit theorems for power variations of fractional type symmetric infinitely divisible random fields. More specifically, the random field X=(X(t))t[0,1]d is defined as an integral of a kernel function g with respect to a symmetric infinitely divisible random measure L and is observed on a grid with mesh size n1. As n, the first order limits are obtained for power variation statistics constructed from rectangular increments of X. The present work is mostly related to [8, 9], who studied a similar problem in the case d=1. We will see, however, that the asymptotic theory in the random field setting is much richer compared to [8, 9] as it contains new limits, which depend on the precise structure of the kernel g. We will give some important examples including the Lévy moving average field, the well-balanced symmetric linear fractional β-stable sheet, and the moving average fractional β-stable field, and discuss potential consequences for statistical inference.

Funding Statement

Vytautė Pilipauskaitė and Mark Podolskij gratefully acknowledge financial support form the project “Ambit fields: Probabilistic properties and statistical inference” funded by Villum Fonden.


The authors are grateful to an anonymous referee for useful comments.


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Andreas Basse-O’Connor. Vytautė Pilipauskaitė. Mark Podolskij. "Power variations for fractional type infinitely divisible random fields." Electron. J. Probab. 26 1 - 35, 2021.


Received: 4 August 2020; Accepted: 29 March 2021; Published: 2021
First available in Project Euclid: 3 May 2021

arXiv: 2008.01412
Digital Object Identifier: 10.1214/21-EJP617

Primary: 60F05, 60G10, 60G22, 60G57, 60G60


Vol.26 • 2021
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