Geometry Of The Expected Value Set And The Set-Valued Sample Mean Process

Abstract

The law of large numbers extends to random sets by employing Minkowski addition. Above that, a central limit theorem is available for set-valued random variables. The existing results use abstract isometries to describe convergence of the sample mean process towards the limit, the expected value set. These statements do not reveal the local geometry and the relations of the sample mean and the expected value set, so these descriptions are not entirely satisfactory in understanding the limiting behavior of the sample mean process. This paper addresses and describes the fluctuations of the sample average mean on the boundary of the expectation set.