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Central limit theorems for empirical product densities of stationary point processes

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  • Lothar Heinrich
  • Stella Klein

Abstract

We prove the asymptotic normality of kernel estimators of second- and higher-order product densities (and of the pair correlation function) for spatially homogeneous (and isotropic) point processes observed on a sampling window $$W_n$$ W n , which is assumed to expand unboundedly in all directions as $$n \rightarrow \infty \,$$ n → ∞ . We first study the asymptotic behavior of the covariances of the empirical product densities under minimal moment and weak dependence assumptions. The proof of the main results is based on the Brillinger-mixing property of the underlying point process and certain smoothness conditions on the higher-order reduced cumulant measures. Finally, the obtained limit theorems enable us to construct $$\chi ^2$$ χ 2 -goodness-of-fit tests for hypothetical product densities. Copyright Springer Science+Business Media Dordrecht 2014

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  • Lothar Heinrich & Stella Klein, 2014. "Central limit theorems for empirical product densities of stationary point processes," Statistical Inference for Stochastic Processes, Springer, vol. 17(2), pages 121-138, July.
    • Handle: RePEc:spr:sistpr:v:17:y:2014:i:2:p:121-138
    DOI: 10.1007/s11203-014-9094-5
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    1. E. Jolivet, 1984. "Upper bound of the speed of convergence of moment density Estimators for stationary point processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 31(1), pages 349-360, December.
    2. A. Baddeley & R. Turner & J. Møller & M. Hazelton, 2005. "Residual analysis for spatial point processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 617-666, November.
    3. Dietrich Stoyan & Helga Stoyan, 2000. "Improving Ratio Estimators of Second Order Point Process Characteristics," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 641-656, December.
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    1. Heinrich, Lothar, 2018. "Brillinger-mixing point processes need not to be ergodic," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 31-35.

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