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Central limit theorems for empirical product densities of stationary point processes. (English) Zbl 1306.60008
Summary: 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 $$W_n$$, which is assumed to expand unboundedly in all directions as $$n \to \infty$$. 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 $$\chi^2$$-goodness-of-fit tests for hypothetical product densities.

##### MSC:
 60F05 Central limit and other weak theorems 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G10 Stationary stochastic processes 62M30 Inference from spatial processes 62G20 Asymptotic properties of nonparametric inference
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##### References:
 [1] Baddeley, A; Turner, R; Møller, J; Hazelton, M, Residual analysis for spatial point processes (with discussion), J R Stat Soc Ser B Stat Methodol, 67, 617-666, (2005) · Zbl 1112.62302 [2] Brillinger DR (1975) Statistical inference for stationary point processes. In: Puri ML (ed) Stochastic Processes and Related Topics, Proceedings of the Summer Research Institute on Statistical Inference for Stochastic Processes, Academic Press, New York, vol 1, 55-99 · Zbl 0666.62032 [3] Cressie NAC (1993) Statistics for spatial data. Wiley, New York [4] Daley DJ, Vere-Jones D (2008) An introduction to the theory of point processes II: genereral theory and structure. Springer, New York · Zbl 1159.60003 [5] Diggle PJ (2003) Statistical analysis of spatial point patterns, 2nd edn. Arnold, London · Zbl 1021.62076 [6] Heinrich, L; Schmidt, V, Normal convergence of multidimensional shot noise and rates of this convergence, Adv Appl Probab, 17, 709-730, (1985) · Zbl 0609.60036 [7] Heinrich, L, Asymptotic gaussianity of some estimators for reduced factorial moment measures and product densities of stationary Poisson cluster processes, Statistics, 19, 87-106, (1988) · Zbl 0666.62032 [8] Heinrich, L; Liebscher, E, Strong convergence of kernel estimators for product densities of absolutely regular point processes, J Nonparam Stat, 8, 65-96, (1997) · Zbl 0884.60041 [9] Heinrich, L; Klein, S, Central limit theorem for the integrated squared error of the empirical second-order product density and goodness-of-fit tests for stationary point processes, Stat Risk Model, 28, 359-387, (2011) · Zbl 1277.60085 [10] Heinrich L (2013) Asymptotic methods in statistics of random point processes. In: Spodarev J (ed) Stochastic geometry, spatial statistics random fields, Lecture notes in mathematics, vol 2068. Springer, New York, pp 115-150 · Zbl 0963.62089 [11] Heinrich, L; Pawlas, Z, Absolute regularity and brillinger-mixing of stationary point processes, Lith Math J, 53, 293-310, (2013) · Zbl 1291.60100 [12] Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Wiley, Chichester · Zbl 1197.62135 [13] Jolivet, E; Bartfai, P (ed.); Tomko, J (ed.), Central limit theorem and convergence of empirical processes for stationary point processes, 117-161, (1981), Amsterdam [14] Jolivet, E, Upper bounds of the speed of convergence of moment density estimators for stationary point processes, Metrika, 31, 349-360, (1984) · Zbl 0576.62055 [15] Karr AF (1986) Point processes and their statistical inference. Marcel Dekker, New York · Zbl 0601.62120 [16] Krickeberg K (1982) Processus ponctuels en statistique. École d’Éte de Probabilités de Saint-Flour X-1980, Lecture notes in mathematics, vol 929. Springer, Berlin, pp 205-313 [17] Leonov, VP; Shiryaev, AN, On a method of calculation of semi-invariants, Theory Probab Appl, 4, 319-329, (1959) · Zbl 0087.33701 [18] Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications, 2nd edn. Wiley, New York · Zbl 0838.60002 [19] Stoyan, D; Stoyan, H, Improving ratio estimators of second order point process characteristics, Scand J Stat, 27, 641-656, (2000) · Zbl 0963.62089
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