On a linear functional for infinitely divisible moving average random fields
Volume 6, Issue 4 (2019), pp. 443–478
Pub. online: 22 October 2019
Type: Research Article
Open Access
Open Access
Received
25 October 2018
25 October 2018
Revised
23 June 2019
23 June 2019
Accepted
13 September 2019
13 September 2019
Published
22 October 2019
22 October 2019
Abstract
Given a low-frequency sample of the infinitely divisible moving average random field $\{{\textstyle\int _{{\mathbb{R}^{d}}}}f(t-x)\Lambda (dx),\hspace{2.5pt}t\in {\mathbb{R}^{d}}\}$, 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 Λ. In this paper, we study asymptotic properties of the linear functional ${L^{2}}(\mathbb{R})\ni v\mapsto {\left\langle v,\widehat{u{v_{0}}}\right\rangle _{{L^{2}}(\mathbb{R})}}$, 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.
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