Central Limit Theorems for Moving Average Random Fields with Non-Random and Random Sampling On Lattices

For a L\'evy basis $L$ on $\mathbb{R}^d$ and a suitable kernel function $f:\mathbb{R}^d \to \mathbb{R}$, consider the continuous spatial moving average field $X=(X_t)_{t\in \mathbb{R}^d}$ defined by $X_t = \int_{\mathbb{R}^d} f(t-s) \, dL(s)$. Based on observations on finite subsets $\Gamma_n$ of $\mathbb{Z}^d$, we obtain central limit theorems for the sample mean and the sample autocovariance function of this process. We allow sequences $(\Gamma_n)$ of deterministic subsets of $\mathbb{Z}^d$ and of random subsets of $\mathbb{Z}^d$. The results generalise existing results for time indexed stochastic processes (i.e. $d=1$) to random fields with arbitrary spatial dimension $d$, and additionally allow for random sampling. The results are applied to obtain a consistent and asymptotically normal estimator of $\mu>0$ in the stochastic partial differential equation $(\mu - \Delta) X = dL$ in dimension 3, where $L$ is L\'evy noise.