# Article

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Keywords:
determinantal point process; permanental point process; trivial tail-$\sigma$-field; exponential moment; shot-noise process; Berry-Esseen bound; multiparameter $K$-function; kernel-type product density estimator; goodness-of-fit test
Summary:
First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function $C(x,y)$ defining an ${\alpha }$-determinantal point process (DPP). Assuming absolute integrability of the function $C_0(x) = C(o,x)$, we show that a stationary ${\alpha }$-DPP with kernel function $C_0(x)$ is strongly'' Brillinger-mixing, implying, among others, that its tail-$\sigma$-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications to statistical second-order analysis of ${\alpha }$-DPPs.
References:
[1] Biscio, C. A. N., Lavancier, F.: Brillinger mixing of determinantal point processes and statistical applications. Electron. J. Stat. (electronic only) 10 582-607 (2016), arXiv: 1507.06506v1 [math ST] (2015). DOI 10.1214/16-EJS1116 | MR 3471989
[2] Camilier, I., Decreusefond, L.: Quasi-invariance and integration by parts for determinantal and permanental processes. J. Funct. Anal. 259 (2010), 268-300. DOI 10.1016/j.jfa.2010.01.007 | MR 2610387 | Zbl 1203.60050
[3] Daley, D. J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods. Probability and Its Applications Springer, New York (2003). MR 1950431 | Zbl 1026.60061
[4] Daley, D. J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure. Probability and Its Applications Springer, New York (2008). MR 2371524 | Zbl 1159.60003
[5] Georgii, H.-O., Yoo, H. J.: Conditional intensity and Gibbsianness of determinantal point processes. J. Stat. Phys. 118 (2005), 55-84. DOI 10.1007/s10955-004-8777-5 | MR 2122549 | Zbl 1130.82016
[6] Heinrich, L.: Asymptotic Gaussianity of some estimators for reduced factorial moment measures and product densities of stationary Poisson cluster processes. Statistics 19 (1988), 87-106. DOI 10.1080/02331888808802075 | MR 0921628 | Zbl 0666.62032
[7] Heinrich, L.: Gaussian limits of empirical multiparameter $K$-functions of homogeneous Poisson processes and tests for complete spatial randomness. Lith. Math. J. 55 (2015), 72-90. DOI 10.1007/s10986-015-9266-z | MR 3323283 | Zbl 1319.60068
[8] Heinrich, L.: On the Brillinger-mixing property of stationary point processes. Submitted (2015), 12 pages.
[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 (2011), 359-387. DOI 10.1524/strm.2011.1094 | MR 2877571 | Zbl 1277.60085
[10] Heinrich, L., Klein, S.: Central limit theorems for empirical product densities of stationary point processes. Stat. Inference Stoch. Process. 17 (2014), 121-138. DOI 10.1007/s11203-014-9094-5 | MR 3219525 | Zbl 1306.60008
[11] Heinrich, L., Prokešová, M.: On estimating the asymptotic variance of stationary point processes. Methodol. Comput. Appl. Probab. 12 (2010), 451-471. DOI 10.1007/s11009-008-9113-3 | MR 2665270 | Zbl 1197.62122
[12] Heinrich, L., Schmidt, V.: Normal convergence of multidimensional shot noise and rates of this convergence. Adv. Appl. Probab. 17 (1985), 709-730. DOI 10.1017/S0001867800015378 | MR 0809427 | Zbl 0609.60036
[13] Hough, J. B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series 51 American Mathematical Society, Providence (2009). MR 2552864 | Zbl 1190.60038
[14] Illian, J., Penttinen, A., Stoyan, H., Stoyan, D.: Statistical Analysis and Modelling of Spatial Point Patterns. Statistics in Practice John Wiley & Sons, Chichester (2008). MR 2384630 | Zbl 1197.62135
[15] Jolivet, E.: Central limit theorem and convergence of empirical processes for stationary point processes. Point Processes and Queuing Problems, Keszthely, 1978 Colloq. Math. Soc. János Bolyai 24 North-Holland, Amsterdam (1981), 117-161. MR 0617406 | Zbl 0474.60040
[16] Karr, A. F.: Estimation of Palm measures of stationary point processes. Probab. Theory Relat. Fields 74 (1987), 55-69. DOI 10.1007/BF01845639 | MR 0863718
[17] Karr, A. F.: Point Processes and Their Statistical Inference. Probability: Pure and Applied 7 Marcel Dekker, New York (1991). MR 1113698 | Zbl 0733.62088
[18] Lavancier, F., Møller, J., Rubak, E.: Determinantal point process models and statistical inference. J. R. Stat. Soc., Ser. B, Stat. Methodol. 77 (2015), 853-877 arXiv: 1205.4818v1-v5 [math ST] (2012-2014). DOI 10.1111/rssb.12096 | MR 3382600
[19] Lenard, A.: States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures. Arch. Rational Mech. Anal. 59 (1975), 241-256. DOI 10.1007/BF00251602 | MR 0391831
[20] Leonov, V. P., Shiryaev, A. N.: On a method of calculation of semi-invariants. Theory Probab. Appl. 4 319-329 (1960), translation from Teor. Veroyatn. Primen. 4 342-355 (1959), Russian 342-355. MR 0123345 | Zbl 0087.33701
[21] Macchi, O.: The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7 (1975), 83-122. DOI 10.1017/S0001867800040313 | MR 0380979 | Zbl 0366.60081
[22] Press, S. J.: Applied Multivariate Analysis: Using Bayesian and Frequentist Methods of Inference. Robert E. Krieger Publishing Company, Malabar (1982). Zbl 0519.62041
[23] Rao, A. R., Bhimasankaram, P.: Linear Algebra. Texts and Readings in Mathematics 19 Hindustan Book Agency, New Delhi (2000). MR 1781860 | Zbl 0982.15001
[24] Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55 923-975 (2000), translation from Usp. Mat. Nauk 55 107-160 (2000), Russian. DOI 10.1070/RM2000v055n05ABEH000321 | MR 1799012 | Zbl 0991.60038
[25] Soshnikov, A.: Gaussian limit for determinantal random point fields. Ann. Probab. 30 (2002), 171-187. DOI 10.1214/aop/1020107764 | MR 1894104 | Zbl 1033.60063
[26] Statulevičius, V. A.: On large deviations. Z. Wahrscheinlichkeitstheorie Verw. Geb. 6 (1966), 133-144. DOI 10.1007/BF00537136 | MR 0221560 | Zbl 0158.36207

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