Power variations for fractional type infinitely divisible random fields

Abstract

This paper presents new limit theorems for power variation of fractional type symmetric infinitely divisible random fields. More specifically, the random field $X=(X(\mathbf{t}){)}_{\mathbf{t}\in [0,1{]}^d}$ 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 $n\to \infty$, the first order limits are obtained for power variation statistics constructed from rectangular increments of $X$. The present work is mostly related to Basse-O'Connor, Lachièze-Rey, Podolskij (2017), Basse-O'Connor, Heinrich, Podolskij (2019), 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 Basse-O'Connor, Lachièze-Rey, Podolskij (2017), Basse-O'Connor, Heinrich, Podolskij (2019) 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 $\beta$-stable sheet, and the moving average fractional $\beta$-stable field, and discuss potential consequences for statistical inference.

References in corpus (1)

• Functional limit theorems for generalized variations of the fractional Brownian sheet

Cited by in corpus (2)

• Proper Scoring Rules for Multivariate Probabilistic Forecasts based on Aggregation and Transformation
• Local scaling limits of Lévy driven fractional random fields