Precise aysmptotics for functionals of point processes having fast decay of correlations

该项目考虑了一个大类的几何泛函的点过程的d维欧氏空间具有快速衰减的相关性。在许多应用中，泛函可以表示为空间相关项（得分函数）的和，它表示点相对于局部有限点集的相互作用。用局部贡献来概括描述有限点集结构的整体几何特征。可能的应用包括单纯复形的统计，高维数据集的统计，胚粒模型的统计和渗流模型的统计。对于泊松和二项式点过程，甚至一般的平方可积泛函被认为是。我们的主要目标是研究这些泛函的精确的大和中等偏差和正常收敛率。我们计划使用和进一步发展的累积量方法（和阶乘矩和集群措施）以及使用和进一步发展Stein的方法.总结我们的目标包括严格的描述和分析的几何泛函的随机结构和几何系统驱动的相关空间结构.

The project considers a large class of geometric functionals of point processes on the d-dimensional Euclidean space having fast decay of correlations. In many application the functionals are expressible as a sum of spatially dependent terms (score functions), which present the interaction of points with respect to locally finite point sets. The sumstypically describe a global geometric feature of a structure of finite point sets in terms of local contributions. Possible applications include statistics of simplicial complexes, statistics of high-dimensional data sets, statistics of germ-grain models and statistics of percolation models. For Poisson and binomial point processes even general square-integrable functionals are considered. Our main goal is to study precise large and moderate deviations and rates of normal convergence for these functionals. We are planing to use and to develop further the method of cumulants (and factorial moments and cluster measures) as well as to use and to develop further Stein’s method.Summarizing our goals comprise the rigorous description and analysis of geometric functionals of random structures and geometric systems driven by correlated spatial structures.