On a linear functional for infinitely divisible moving average random fields

Abstract

Given a low-frequency sample of the infinitely divisible moving average random field $\{{\int }_{{\mathbb{R}}^d}f(t-x)\Lambda (dx),t\in {\mathbb{R}}^d\}$, in [13] we proposed an estimator $\widehat{u{v}_0}$ for the function $\mathbb{R}\ni x\mapsto u(x)v_0(x)=(u{v}_0)(x)$, with $u(x)=x$ and $v_0$ being the Lévy density of the integrator random measure $\Lambda$. In this paper, we study asymptotic properties of the linear functional $L^2(\mathbb{R})\ni v\mapsto {⟨v,\widehat{u{v}_0}⟩}_{L^2(\mathbb{R})}$, if the (known) kernel function $f$ has a compact support. We provide conditions that ensure consistency (in mean) and prove a central limit theorem for it.

Published at https://doi.org/10.15559/19-VMSTA143 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/)