Central limit theorems for poisson hyperplane tessellations

Annals of Applied Probability

Volumn 16, Issue 2, 2006, Pages 919-950

  Heinrich, Lothar ^a   Schmidt, Hendrik ^b   Schmidt, Volker ^b  

^a UNIVERSITY OF AUGSBURG   (Germany)

^b UNIVERSITY OF ULM   (Germany)

Author keywords

Central limit theorem; Confidence interval; Hoeffding's decomposition; K flat intersection process; Long range dependence; Point process; Poisson hyperplane process; U statistic

Indexed keywords

EID: 33746907869     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051606000000033     Document Type: Article

References (32)

1. AVRAM, F. and BERTSIMAS, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 1033-1046. MR1241033
2. BADDELEY, A. J. (1980). A limit theorem for statistics of spatial data. Adv. in Appl. Probab. 12 447-461. MR0569436
3. BICKEL, P. and DOKSUM, K. (2001 ). Mathematical Statistics: Basic Ideas and Selected Topics 1, 2nded. Prentice Hall, London. MR0443141
4. BEIL, M., ECKEL, S., FLEISCHER, F., SCHMIDT, H., SCHMIDT, V. and WALTHER, P. (2006). Fitting of random tessellation models to cytoskeleton network data. J. Theoret. Biol. To appear.
5. BÖHM, S., HEINRICH, L. and SCHMIDT, V. (2004). Asymptotic properties of estimators for the volume fractions of jointly stationary random sets. Statist. Neerlandica 58 388-406. MR2106346
6. DALEY, D. J. and VERE-JONES, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York. MR0950166
7. GLOAGUEN, C., COUPÉ, P., MAIER, R. and SCHMIDT, V. (2002). Stochastic modelling of urban access networks. In Proc. 10th Internat. Telecommun. Network Strategy Planning Symp. 99-104. VDE, Berlin.
8. GLOAGUEN, C., FLEISCHER, F., SCHMIDT, H. and SCHMIDT, V. (2006). Fitting of stochastic telecommunication network models, via distance measures and Monte Carlo tests. In Telecommunication Systems. To appear.
9. HEINRICH, L. (1994). Normal approximation for some mean-value estimates of absolutely regular tessellations. Math. Methods Statist. 3 1-24. MR1272628
10. HEINRICH, L., KÖRNER, R., MEHLHORN, N. and MUCHE, L. (1998). Numerical and analytical computation of some second-order characteristics of spatial Poisson-Voronoi tessellations. Statistics 31 235-259. MR1718316
11. HEINRICH, L. and MOLCHANOV, I. S. (1999). Central limit theorem for a class of random measures associated with germ-grain models. Adv. in Appl. Probab. 31 283-314. MR1724553
12. HEINRICH, L. and MUCHE, L. (2007). Second-order properties of the point process of nodes in a stationary Voronoi tessellation. Math. Nachr. To appear.
13. HEINRICH, L., SCHMIDT, H. and SCHMIDT, V. (2005). Limit theorems for stationary tessellations with random inner cell structures. Adv. in Appl. Probab. 37 25-47. MR2135152
14. KARR, A. F. (1993). Probability. Springer, New York. MR1231974
15. KOROLJUK, V. S. and BOROVSKICH, YU. V. (1994). Theory of U-Statistics. Kluwer, Dordrecht. MR1472486
16. MASE, S. (1982). Asymptotic properties of stereological estimators of the volume fraction for stationary random sets. J. Appl. Probab. 19 111-126. MR0644424
17. MATHERON, G. (1975). Random Sets and Integral Geometry. Wiley, New York. MR0385969
18. MOLCHANOV, I. S. (1995). Statistics for the Boolean model: From the estimation of means to the estimation of distributions. Adv. in Appl. Probab. 27 63-86. MR1315578
19. MOLCHANOV, I. S. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester. Zbl. 0878.62068.
20. d. Adv. in Appl. Probab. 21 37-73. MR0980736
21. MØLLER, J. (1994). Lectures on Random Voronoi Tessellations. Springer, New York. MR1295245
22. OKABE, A., BOOTS, B., SUGIHARA, K. and CHIU, S. N. (2000). Spatial Tessellations, 2nd ed. Wiley, Chichester. MR1770006
23. PANTLE, U., SCHMIDT, V. and SPODAREV, E. (2006). Central limit theorems for functionals of stationary germ-grain models. Adv. in Appl. Probab. 38 76-94.
24. PAROUX, K. (1998). Quelques théorèmes centraux limites pour les processus poissoniens de droites dans le plan. Adv. in Appl. Probab. 30 640-656. MR1663517
25. PENROSE, M. D. and YUKICH, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005-1041. MR1878288
26. RÉNYI, A. (1970). Probability Theory. North-Holland, Amsterdam. MR0315747
27. SCHLADITZ, K. (2000). Estimation of the intensity of stationary flat processes. Adv. in Appl. Probab. 32 114-139. MR1765167
28. SCHMIDT, V. and SPODAREV, E. (2005). Joint estimators for the specific intrinsic volumes of stationary random sets. Stochastic Process. Appl. 115 959-981. MR2138810
29. SCHNEIDER, R. and WEIL, W. (1992). Integralgeometrie. Teubner, Stuttgart. MRI203777
30. [30[ SCHNEIDER, R. and WEIL. W. (2000). Stochastische Geometrie. Teubner. Stuttgart. MR1794753
31. SERFLING, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York. MR0595165
32. STOYAN, D., KENDALL, W. S. and MECKE, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Wiley, Chichester. MR0895588