Limit theorems for random compact sets in Banach space

Abstract

The concept of random set, though vaguely known for a long time (possibly since Buffon's needle problem), did not develop until Robbins [25, 26] provided for the first time a solid mathematical formulation of this concept and investigated relationships between random sets and geometric probabilities. Later on (in a different context) this concept gave rise to a more general concept of set-valued function in topology, and applications were also found in several areas such as economics (see, for example, Aumann[3] and Debreu[10]) and control theory (see, for example, Hermes [13]), among others. Recently in two independent formulations, D. G. Kendall [17] and Matheron[21] provided a comprehensive mathematical theory of this concept influenced by the geometric probability point of view. Actually, much of the research in this area falls under the heading of stochastic geometry (see, for example, M. G. Kendall and P. A. P. Moran [18]). The D. G. Kendall and Matheron theories have been compared and ‘reconciled’ by Ripley [24]. In the past few years random sets have been investigated as extensions of random variables and random vectors, and in this framework the problems of deriving limit theorems have received a great deal of attention. This approach is benefiting greatly from probability results in Banach spaces.