arXiv 1902.012552019-02math.PR

格上非随机和随机抽样滑动平均随机场的中心极限定理

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

David Berger

摘要：

对于$\mathbb{R}^d$上的Lévy基$L$和合适的核函数$f:\mathbb{R}^d \to \mathbb{R}$，考虑由$X_t = \int_{\mathbb{R}^d} f(t-s) \, dL(s)$定义的连续空间移动平均场$X=(X_t)_{t\in \mathbb{R}^d}$。基于对$\mathbb{Z}^d$的有限子集$Γ_n$的观测，我们得到了该过程的样本均值和样本自协方差函数的中心极限定理。我们允许序列$(Γ_n)$的确定性子集的$\mathbb{Z}^d$和随机子集的$\mathbb{Z}^d$。结果概括了现有的时间索引随机过程（即$d=1$）的随机场与任意空间维度$d$的结果，并允许随机抽样。应用所得结果得到了3维随机偏微分方程$(μ- Δ) X = dL$中$μ>0$的相合渐近正态估计，其中$L$为Lévy噪声.

英文摘要：

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 $Γ_n$ of $\mathbb{Z}^d$, we obtain central limit theorems for the sample mean and the sample autocovariance function of this process. We allow sequences $(Γ_n)$ 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 $μ>0$ in the stochastic partial differential equation $(μ- Δ) X = dL$ in dimension 3, where $L$ is Lévy noise.

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摘要：

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