On a linear functional for infinitely divisible moving average random fields
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Given a low-frequency sample of the infinitely divisible moving average random field {∫_R^d f(t-x) Λ(dx), t ∈R^d }, in [1], we proposed an estimator uv_0 for the function R∋ x u(x)v_0(x) = (uv_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(R) ∋ v 〈 v, uv_0〉_L^2(R), if the (known) kernel function f has compact support. We provide conditions that ensure consistency (in mean) and prove a central limit theorem for it.
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