Power variations for fractional type infinitely divisible random fields

Andreas Basse-O'Connor, Vytaute Pilipauskaite, Mark Podolskij

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Abstract

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 n −1 . 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.

Original languageEnglish
Article number55
JournalElectronic Journal of Probability
Volume26
Number of pages36
ISSN1083-6489
DOIs
Publication statusPublished - May 2021

Keywords

  • Fractional fields
  • Infill asymptotics
  • Limit theorems
  • Moving averages
  • Power variation
  • Stable convergence

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