Power variations for fractional type infinitely divisible random fields (2008.01412v3)

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.