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

Abstract

In this paper we present some limit theorems for power variation of a class of stationary increments Lévy driven moving averages in the setting of critical regimes. In an earlier work the authors derived first and second order asymptotic results for kth order increments of stationary increments Lévy driven moving averages. 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 determined by the power . In this work we study the critical cases and with , which were not covered in the above mentioned work.

Keywords:

• Power variation
• limit theorems
• moving averages
• fractional processes
• stable convergence
• high frequency data

Acknowledgements

We would like to thank the anonymous referee for careful reading of the manuscript and helpful suggestions. Andreas Basse-O’Connor’s research was supported by the grant DFF–4002-00003 from the Danish Council for Independent Research. Mark Podolskij gratefully acknowledges financial support through the research project “Ambit fields: probabilistic properties and statistical inference” funded by Villum Fonden.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Andreas Basse-O’Connor’s research was supported by the grant DFF–4002-00003 from the Danish Council for Independent Research. Mark Podolskij gratefully acknowledges financial support through the research project “Ambit fields: probabilistic properties and statistical inference” funded by Villum Fonden.