| Title:
|
Gaussian approximation for functionals of Gibbs particle processes (English) |
| Author:
|
Flimmel, Daniela |
| Author:
|
Beneš, Viktor |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
54 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
765-777 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the paper asymptotic properties of functionals of stationary Gibbs particle processes are derived. Two known techniques from the point process theory in the Euclidean space $\mathbb{R}^d$ are extended to the space of compact sets on $\mathbb{R}^d$ equipped with the Hausdorff metric. First, conditions for the existence of the stationary Gibbs point process with given conditional intensity have been simplified recently. Secondly, the Malliavin-Stein method was applied to the estimation of Wasserstein distance between the Gibbs input and standard Gaussian distribution. We transform these theories to the space of compact sets and use them to derive a Gaussian approximation for functionals of a planar Gibbs segment process. (English) |
| Keyword:
|
asymptotics of functionals |
| Keyword:
|
innovation |
| Keyword:
|
stationary Gibbs particle process |
| Keyword:
|
Wasserstein distance |
| MSC:
|
60D05 |
| MSC:
|
60G55 |
| idZBL:
|
Zbl 06987033 |
| idMR:
|
MR3863255 |
| DOI:
|
10.14736/kyb-2018-4-0765 |
| . |
| Date available:
|
2018-10-30T14:49:56Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147423 |
| . |
| Reference:
|
[1] Beneš, V., Večeřa, J., Pultar, M.: Planar segment processes with reference mark distributions, modeling and simulation..Methodol. Comput. Appl. Probab. (2018), accepted. 10.1007/s11009-017-9608-x |
| Reference:
|
[2] Blaszczyszyn, B., Yogeshwaran, D., Yukich, J. E.: Limit theory for geometric statistics of point processes having fast decay of correlations..Preprint (2018), submitted to the Annals of Probab. |
| Reference:
|
[3] Daley, D. J., Vere-Jones, D.: An Introduction to the Theory of Point Processes..Volume I: Elementary Theory and Methods. MR 1950431 |
| Reference:
|
[4] Dereudre, D.: Introduction to the theory of Gibbs point processes..Preprint (2017), submitted. |
| Reference:
|
[5] Georgii, H.-O.: Gibbs Measures and Phase Transitions. Second edition..W. de Gruyter and Co., Berlin 2011. MR 2807681, 10.1515/9783110250329 |
| Reference:
|
[6] Last, G., Penrose, M.: Lectures on the Poisson Process..Cambridge University Press, Cambridge 2017. MR 3791470, 10.1017/9781316104477 |
| Reference:
|
[7] Mase, S.: Marked Gibbs processes and asymptotic normality of maximum pseudo-likelihood estimators..Math. Nachr. 209 (2000), 151-169. MR 1734363, 10.1002/(sici)1522-2616(200001)209:1<151::aid-mana151>3.0.co;2-j |
| Reference:
|
[8] Ruelle, D.: Superstable interactions in classical statistical mechanics..Commun. Math. Phys. 18 (1970), 127-159. MR 0266565, 10.1007/bf01646091 |
| Reference:
|
[9] Schneider, R., Weil, W.: Stochastic and Integral Geometry..Springer, Berlin 2008. Zbl 1175.60003, MR 2455326, 10.1007/978-3-540-78859-1 |
| Reference:
|
[10] Schreiber, T., Yukich, J. E.: Limit theorems for geometric functionals of Gibbs point processes..Ann. Inst. Henri Poincaré - Probab. et Statist. 49 (2013), 1158-1182. MR 3127918, 10.1214/12-aihp500 |
| Reference:
|
[11] Serra, J.: Image Analysis and Mathematical Morphology..Academic Press, London 1982. MR 0753649, 10.1002/cyto.990040213 |
| Reference:
|
[12] Stucki, K., Schuhmacher, D.: Bounds for the probability generating functional of a Gibbs point process..Adv. Appl. Probab. 46 (2014), 21-34. MR 3189046, 10.1239/aap/1396360101 |
| Reference:
|
[13] Torrisi, G. L.: Probability approximation of point processes with Papangelou conditional intensity..Bernoulli 23 (2017), 2210-2256. MR 3648030, 10.3150/16-bej808 |
| Reference:
|
[14] Večeřa, J., Beneš, V.: Approaches to asymptotics for U-statistics of Gibbs facet processes..Statist. Probab. Let. 122 (2017), 51-57. MR 3584137, 10.1016/j.spl.2016.10.024 |
| Reference:
|
[15] Xia, A., Yukich, J. E.: Normal approximation for statistics of Gibbsian input in geometric probability..Adv. Appl. Probab. 25 (2015), 934-972. MR 3433291, 10.1017/s0001867800048965 |
| . |
