  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.
