On the strong Brillinger-mixing property of α -determinantal point processes and some applications

Lothar Heinrich

Applications of Mathematics (2016)

  • Volume: 61, Issue: 4, page 443-461
  • ISSN: 0862-7940

Abstract

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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 α -determinantal point process (DPP). Assuming absolute integrability of the function C 0 ( x ) = C ( o , x ) , we show that a stationary α -DPP with kernel function C 0 ( x ) is “strongly” Brillinger-mixing, implying, among others, that its tail- σ -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 α -DPPs.

How to cite

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Heinrich, Lothar. "On the strong Brillinger-mixing property of ${\alpha }$-determinantal point processes and some applications." Applications of Mathematics 61.4 (2016): 443-461. <http://eudml.org/doc/283407>.

@article{Heinrich2016,
abstract = {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.},
author = {Heinrich, Lothar},
journal = {Applications of Mathematics},
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; 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},
language = {eng},
number = {4},
pages = {443-461},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the strong Brillinger-mixing property of $\{\alpha \}$-determinantal point processes and some applications},
url = {http://eudml.org/doc/283407},
volume = {61},
year = {2016},
}

TY - JOUR
AU - Heinrich, Lothar
TI - On the strong Brillinger-mixing property of ${\alpha }$-determinantal point processes and some applications
JO - Applications of Mathematics
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 4
SP - 443
EP - 461
AB - 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.
LA - eng
KW - 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; 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
UR - http://eudml.org/doc/283407
ER -

References

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  1. Biscio, C. A. N., Lavancier, F., 10.1214/16-EJS1116, Electron. J. Stat. (electronic only) 10 582-607 (2016), arXiv: 1507.06506v1 [math ST] (2015). (2015) MR3471989DOI10.1214/16-EJS1116
  2. Camilier, I., Decreusefond, L., 10.1016/j.jfa.2010.01.007, J. Funct. Anal. 259 (2010), 268-300. (2010) Zbl1203.60050MR2610387DOI10.1016/j.jfa.2010.01.007
  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). (2003) Zbl1026.60061MR1950431
  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). (2008) Zbl1159.60003MR2371524
  5. Georgii, H.-O., Yoo, H. J., 10.1007/s10955-004-8777-5, J. Stat. Phys. 118 (2005), 55-84. (2005) Zbl1130.82016MR2122549DOI10.1007/s10955-004-8777-5
  6. Heinrich, L., 10.1080/02331888808802075, Statistics 19 (1988), 87-106. (1988) Zbl0666.62032MR0921628DOI10.1080/02331888808802075
  7. Heinrich, L., 10.1007/s10986-015-9266-z, Lith. Math. J. 55 (2015), 72-90. (2015) Zbl1319.60068MR3323283DOI10.1007/s10986-015-9266-z
  8. Heinrich, L., On the Brillinger-mixing property of stationary point processes, Submitted (2015), 12 pages. 
  9. Heinrich, L., Klein, S., 10.1524/strm.2011.1094, Stat. Risk Model. 28 (2011), 359-387. (2011) Zbl1277.60085MR2877571DOI10.1524/strm.2011.1094
  10. Heinrich, L., Klein, S., 10.1007/s11203-014-9094-5, Stat. Inference Stoch. Process. 17 (2014), 121-138. (2014) Zbl1306.60008MR3219525DOI10.1007/s11203-014-9094-5
  11. Heinrich, L., Prokešová, M., 10.1007/s11009-008-9113-3, Methodol. Comput. Appl. Probab. 12 (2010), 451-471. (2010) Zbl1197.62122MR2665270DOI10.1007/s11009-008-9113-3
  12. Heinrich, L., Schmidt, V., 10.1017/S0001867800015378, Adv. Appl. Probab. 17 (1985), 709-730. (1985) Zbl0609.60036MR0809427DOI10.1017/S0001867800015378
  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). (2009) Zbl1190.60038MR2552864
  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). (2008) Zbl1197.62135MR2384630
  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. (1981) Zbl0474.60040MR0617406
  16. Karr, A. F., 10.1007/BF01845639, Probab. Theory Relat. Fields 74 (1987), 55-69. (1987) MR0863718DOI10.1007/BF01845639
  17. Karr, A. F., Point Processes and Their Statistical Inference, Probability: Pure and Applied 7 Marcel Dekker, New York (1991). (1991) Zbl0733.62088MR1113698
  18. Lavancier, F., Møller, J., Rubak, E., 10.1111/rssb.12096, J. R. Stat. Soc., Ser. B, Stat. Methodol. 77 (2015), 853-877 arXiv: 1205.4818v1-v5 [math ST] (2012-2014). (2012) MR3382600DOI10.1111/rssb.12096
  19. Lenard, A., 10.1007/BF00251602, Arch. Rational Mech. Anal. 59 (1975), 241-256. (1975) MR0391831DOI10.1007/BF00251602
  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. (1959) Zbl0087.33701MR0123345
  21. Macchi, O., 10.1017/S0001867800040313, Adv. Appl. Probab. 7 (1975), 83-122. (1975) Zbl0366.60081MR0380979DOI10.1017/S0001867800040313
  22. Press, S. J., Applied Multivariate Analysis: Using Bayesian and Frequentist Methods of Inference, Robert E. Krieger Publishing Company, Malabar (1982). (1982) Zbl0519.62041
  23. Rao, A. R., Bhimasankaram, P., Linear Algebra, Texts and Readings in Mathematics 19 Hindustan Book Agency, New Delhi (2000). (2000) Zbl0982.15001MR1781860
  24. Soshnikov, A., 10.1070/RM2000v055n05ABEH000321, Russ. Math. Surv. 55 923-975 (2000), translation from Usp. Mat. Nauk 55 107-160 (2000), Russian. (2000) Zbl0991.60038MR1799012DOI10.1070/RM2000v055n05ABEH000321
  25. Soshnikov, A., 10.1214/aop/1020107764, Ann. Probab. 30 (2002), 171-187. (2002) Zbl1033.60063MR1894104DOI10.1214/aop/1020107764
  26. Statulevičius, V. A., 10.1007/BF00537136, Z. Wahrscheinlichkeitstheorie Verw. Geb. 6 (1966), 133-144. (1966) Zbl0158.36207MR0221560DOI10.1007/BF00537136

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