Power variations for fractional type infinitely divisible random fields

TL;DR

This paper develops new limit theorems for power variation of fractional symmetric infinitely divisible random fields, revealing richer asymptotic behavior in higher dimensions and providing examples relevant for statistical inference.

Contribution

It extends previous one-dimensional results to higher-dimensional random fields, uncovering new limit behaviors depending on the kernel structure.

Findings

New limit theorems for power variation in random fields

Identification of limits depending on kernel structure

Applications to Lévý moving average and fractional stable fields

Abstract

This paper presents new limit theorems for power variation of fractional type symmetric infinitely divisible random fields. More specifically, the random field $X = (X(\boldsymbol{t}))_{\boldsymbol{t} \in [0,1]^d}$ 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, Lachi\`eze-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, Lachi\`eze-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 L\'evy 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.