标题： 关于随机闭集均值密度的局部逼近 显示英文标题

标题： On the local approximation of mean densities of random closed sets

摘要： 低维随机闭集的平均密度以及全维随机集的平均边界密度及其估计，在许多实际应用中具有重要意义。 迄今为止，当前文献中仅有一些部分结果可用，前提是随机集要么是平稳的，要么是布尔模型，要么其颗粒是凸的。 我们在此考虑非平稳的随机闭集（不一定为布尔模型），其颗粒需满足某些一般的正则性条件，从而推广了先前的结果。 我们解决了（Bernoulli 15 (2009) 1222-1242）中提出的关于通过点态极限逼近低维随机集平均密度的问题，以及Matheron在(Random Sets and Integral Geometry (1975) Wiley)中提出的关于全维随机闭集所谓的特定面积的存在性（及其值）的问题。 此外，还讨论了与球面接触分布函数之间的关系，以及一些例子和应用。 显示英文摘要

摘要： Mean density of lower dimensional random closed sets, as well as the mean boundary density of full dimensional random sets, and their estimation are of great interest in many real applications. Only partial results are available so far in current literature, under the assumption that the random set is either stationary, or it is a Boolean model, or it has convex grains. We consider here non-stationary random closed sets (not necessarily Boolean models), whose grains have to satisfy some general regularity conditions, extending previous results. We address the open problem posed in (Bernoulli 15 (2009) 1222-1242) about the approximation of the mean density of lower dimensional random sets by a pointwise limit, and to the open problem posed by Matheron in (Random Sets and Integral Geometry (1975) Wiley) about the existence (and its value) of the so-called specific area of full dimensional random closed sets. The relationship with the spherical contact distribution function, as well as some examples and applications are also discussed.