In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

In the present paper, we propose a Palm likelihood approach as a general estimating principle for stationary point processes in R d for which the density of the second-order factorial moment measure is available in closed form. Examples of such point processes include the Neyman-Scott processes and the log Gaussian Cox processes. The computations involved in determining the Palm likelihood estimator are simple. Conditions are provided under which the Palm likelihood estimator is consistent and asymptotically normally distributed. R 0 [ĝ c (u) − g c (u; θ)] 2 du

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We investigate a class of kernel estimators σ 2 n of the asymptotic variance σ 2 of a d-dimensional stationary point process Ψ = i≥1 δ X i which can be observed in a cubic sampling windowand its existence is guaranteed whenever the corresponding reduced covariance measure γ (2) red (·) has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of γ (2) red (·) outside of an expanding ball centered at the origin, we determine optimal bandwidths b n (up to a constant) minimizing the mean squared error of σ 2 n . The case when γ (2) red (·) has bounded support is of particular interest. Further we suggest an isotropised estimator σ 2 n suitable for motion-invariant point processes and compare its properties with σ 2 . Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of σ 2 for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b n .

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.