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

Abstract

For a Lévy 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 Lévy noise.

References in corpus (1)

• Limit theorems for the sample autocovariance of a continuous-time moving average process with long memory

Cited by in corpus (1)

• Estimation of causal CARMA random fields