A new definition of random set

Abstract

A new definition of random sets is proposed in the presented paper. It is based on a special distance in a measurable space and uses negative definite kernels for continuation from the initial space to the one of the random sets. Motivation for introducing the new definition is that the classical approach deals with Hausdorff distance between realisations of the random sets, which is not satisfactory for statistical analysis in many cases. We place the realisations of the random sets in a complete Boolean algebra (B.A.) endowed with a positive finite measure intended to capture important characteristics of the realisations. A distance on B.A. is introduced as a square root of measure of symmetric difference between its two elements. The distance is then used to define a class of Borel subsets of B.A. Consequently, random sets are defined as measurable mappings taking values in the B.A. This approach enables us to use more general family of distances between realisations of random sets which allows us to make new statistical tests concerning equality of some characteristics of random set distributions. As an extra result, the notion of stability of newly defined random sets with respect to intersections is proposed and limit theorems are obtained.