On a linear functional for infinitely divisible moving average random fields

TL;DR

This paper investigates the asymptotic behavior of a linear functional estimator for the Lévy density in infinitely divisible moving average random fields, establishing consistency and a central limit theorem under certain conditions.

Contribution

It extends previous work by analyzing the asymptotic properties of a linear functional estimator with known kernel support, including consistency and distributional convergence.

Findings

Established mean consistency of the linear functional estimator.

Proved a central limit theorem for the estimator.

Provided conditions on the kernel function for asymptotic properties.

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 $\hat{uv_0}$ for the function $\mathbb{R}\ni x\mapsto u(x)v_0(x)=(uv_0)(x)$, with $u(x)=x$ and $v_0$ being the L\'{e}vy density of the integrator random measure $\Lambda$. In this paper, we study asymptotic properties of the linear functional $L^2(\mathbb{R})\ni v\mapsto \left \langle v,\hat{uv_0}\right \rangle_{L^2(\mathbb{R})}$, 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.