## 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 language | English |
---|---|

Article number | 55 |

Journal | Electronic Journal of Probability |

Volume | 26 |

Number of pages | 36 |

ISSN | 1083-6489 |

DOIs | |

Publication status | Published - May 2021 |

## Keywords

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