A Strong Law of Large Numbers for Random Compact Sets

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. A strong law of large numbers is shown for random sets taking values in the nonempty, compact subsets of Rh. 1. Introduction. In the study of probabilities on geometrical objects, there have been some recent attempts to formulate general theories of random sets, notably by Kendall [8] and Matheron [10]. It is our purpose here to make a contribution in this direction by demonstrating the existence of a strong law of large numbers for random sets taking values in the class of compact subsets of RE. The result is proved first under the assumption of convexity and then extended to the general case. In the spirit of previous (nonprobabilistic) work by Castaing [6], Debreu [7], and Rockafellar [12] among others, we find it use-ful to proceed from the definition of a random set as a measurable set-valued function. Our particular concern with the behavior of sums of random sets arose in the formulation of a stochastic model of growth which seems useful in certain ap-plications where enlargement occurs by surface accretion. Mathematically it is appealing to model such a dynamic by the set addition of random "growth ele-ments." The question of asymptotic shape then leads naturally to the considera-tion of normalized sums of random sets.