For a stationary moving average random field, a nonparametric low frequency estimator of the Lévy density of its infinitely divisible independently scattered integrator measure is given. The plug-in estimate is based on the solution w of the linear integral equation $v(x)={\int }_{{ℝ}^d}g(s)w(h(s)x)ds$, where $g,h:{ℝ}^d\to ℝ$ are given measurable functions and v is a (weighted) $L^2$-function on $ℝ$. We investigate conditions for the existence and uniqueness of this solution and give $L^2$-error bounds for the resulting estimates. An application to pure jump moving averages and a simulation study round off the paper.

中文翻译：

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一种线性积分方程的解，适用于无限可分移动平均线的统计

对于静止的移动平均随机场，给出了其无限可分独立散射积分器测度的 Lévy 密度的非参数低频估计量。插件估计是基于线性积分方程的解w$v(X)={\int }_{{ℝ}^d}G(s)w(H(s)X)ds$， 在哪里$G,H：{ℝ}^d\to ℝ$给定可测量的函数，并且v是（加权的）${\mathrm{大号}}^2$- 功能开启$ℝ$. 我们研究这个解的存在和唯一性的条件，并给出${\mathrm{大号}}^2$- 结果估计的误差范围。对纯跳跃移动平均线的应用和模拟研究使本文更加完善。