Moderate deviations for some point measures in geometric probability

Baryshnikov, Yu; Eichelsbacher, P.; Schreiber, T.; Yukich, J. E.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 3, pp. 422-446.

Les fonctionnelles en probabilite géométrique s’expriment souvent comme des sommes de fonctions bornées qui possèdent la fonction de stabilisation. Les méthodes de cumulants et les modifications exponentielles des mesures démontrent que ces fonctionnelles vérifient le principe des déviations modérées. Ceci donne des principes des déviations modérées et des lois de logarithme itéré pour des modèles de ‘packing aléatoires’ ainsi que pour des statistiques de modèles de ‘germe-grain’ et de graphes avec $k$ plus proches voisins.

Functionals in geometric probability are often expressed as sums of bounded functions exhibiting exponential stabilization. Methods based on cumulant techniques and exponential modifications of measures show that such functionals satisfy moderate deviation principles. This leads to moderate deviation principles and laws of the iterated logarithm for random packing models as well as for statistics associated with germ-grain models and $k$ nearest neighbor graphs.

Classification : 60F05, 60D05
Mots clés : moderate deviations, laws of the iterated logarithm, random euclidean graphs, random sequential packing

[1] A. De Acosta. Exponential tightness and projective systems in large deviation theory. In Festschrift for Lucien Le Cam 143-156. Springer, New York, 1997. | MR | Zbl

[2] A. D. Barbour and A. Xia. Normal approximation for random sums. Adv. in Appl. Probab. 38 (2006) 693-728. | MR | Zbl

[3] Y. Baryshnikov and J. E. Yukich. Gaussian fields and random packing. J. Statist. Phys. 111 (2003) 443-463. | MR | Zbl

[4] Y. Baryshnikov and J. E. Yukich. Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005) 213-253. | MR | Zbl

[5] S. N. Chiu and M. P. Quine. Central limit theory for the number of seeds in a growth model in ℝd with inhomogeneous Poisson arrivals. Ann. Appl. Probab. 7 (1997) 802-814. | MR | Zbl

[6] S. N. Chiu and M. P. Quine. Central limit theorem for germination-growth models in ℝd with non-Poisson locations. Adv. in Appl. Probab. 33 (2001) 751-755. | MR | Zbl

[7] E. G. Coffman, L. Flatto, P. Jelenković and B. Poonen. Packing random intervals on-line. Algorithmica 22 (1998) 448-476. | MR | Zbl

[8] A. Dvoretzky and H. Robbins. On the “parking” problem. MTA Mat. Kut. Int. Kzl. (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences) 9 (1964) 209-225. | MR | Zbl

[9] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Springer, New York, 1988. | MR | Zbl

[10] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | MR | Zbl

[11] J.-D. Deuschel and D. Stroock. Large Deviations. Academic Press, Boston, MA, 1989. | MR | Zbl

[12] P. Eichelsbacher and U. Schmock. Large deviations for products of empirical measures in strong topologies and applications. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 779-797. | Numdam | MR | Zbl

[13] P. Eichelsbacher and U. Schmock. Rank-dependent moderate deviations of U-empirical measures in strong topologies. Probab. Theory Related Fields 126 (2003) 61-90. | MR | Zbl

[14] R. Fernández, P. Ferrari and N. Garcia. Measures on contour, polymer or animal models. A probabilistic approach. Markov Process. Related Fields 4 (1998) 479-497. | MR | Zbl

[15] R. Fernández, P. Ferrari and N. Garcia. Loss network representation of Ising contours. Ann. Probab. 29 (2001) 902-937. | MR | Zbl

[16] R. Fernández, P. Ferrari and N. Garcia. Perfect simulation for interacting point processes, loss networks and Ising models. Stochastic Process Appl. 102 (2002) 63-88. | MR | Zbl

[17] G. Grimmett. Percolation. Grundlehren der mathematischen Wissenschaften 321, Springer, Berlin, 1999. | MR | Zbl

[18] P. Hall. Introduction to the Theory of Coverage Processes. Wiley, New York, 1988. | MR | Zbl

[19] L. Heinrich and I. Molchanov. Central limit theorem for a class of random measures associated with germ-grain models. Adv. in Appl. Probab. 31 (1999) 283-314. | MR | Zbl

[20] M. N. M. Van Lieshout and R. Stoica. Perfect simulation for marked point processes. Compt. Statist. Data Anal. 51 (2006) 679-698. | MR | Zbl

[21] T. M. Liggett, R. H. Schonmann and A. M. Stacey. Domination by product measures. Ann. Probab. 25 (1997) 71-95. | MR | Zbl

[22] V. A. Malyshev and R. A. Minlos. Gibbs Random Fields. Kluwer, Dordrecht, 1991. | MR | Zbl

[23] M. D. Penrose. Random Geometric Graphs. Clarendon Press, Oxford, 2003. | MR | Zbl

[24] M. D. Penrose. Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33 (2005) 1945-1991. | MR | Zbl

[25] M. D. Penrose. Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 (2007) 1124-1150. | MR | Zbl

[26] M. D. Penrose. Gaussian limits for random geometric measures. Electronic J. Probab. 12 (2007) 989-1035. | MR | Zbl

[27] M. D. Penrose and J. E. Yukich. Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001) 1005-1041. | MR | Zbl

[28] M. D. Penrose and J. E. Yukich. Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002) 272-301. | MR | Zbl

[29] M. D. Penrose and J. E. Yukich. Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003) 277-303. | MR | Zbl

[30] M. D. Penrose and J. E. Yukich. Normal approximation in geometric probability. In Stein's Method and Applications. A. D. Barbour and Louis H. Y. Chen (Eds) 37-58. Institute for Mathematical Sciences, National University of Singapore, 2005. Available at http://www.lehigh.edu/~jey0/publications.html. | MR

[31] A. Rényi, Théorie des éléments saillants d'une suite d'observations. In Colloquium on Combinatorial Methods in Probability Theory 104-115. Mathematical Institut, Aarhus Universitet, Denmark, 1962. | Zbl

[32] L. Saulis and V. Statulevicius. Limit theorems on large deviations. In: Limit Theorems of Probability Theory. Y. V. Prokhorov and V. Statulevicius (Eds). Springer, 2000. | Zbl

[33] T. Schreiber and J. E. Yukich. Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points. Ann. Probab. 36 (2008) 363-396. | MR | Zbl

[34] D. Stoyan, W. Kendall and J. Mecke. Stochastic Geometry and Its Applications, 2nd edition. Wiley, Chichester, 1995. | MR | Zbl