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

2019/02/04 by Berger, David
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1902.01255

Abstract

For a Lévy basis L on ℝ^d and a suitable kernel function f:ℝ^d → ℝ, consider the continuous spatial moving average field X=(X_t)_{t∈ ℝ^d} defined by X_t = ∫_ℝ^d f(t-s) dL(s). Based on observations on finite subsets Γ_n of 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 ℤ^d and of random subsets of 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.

Related

• A central limit theorem for stationary random fields
• Extremes of Lévy-driven spatial random fields with regularly varying Lévy measure
• A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process
• A strong central limit theorem for a class of random surfaces
• Central limit theorem for Fourier transform and periodogram of random fields