A Strong Law of Large Numbers for Random Compact Sets

Abstract. The objective of this paper is to develop a set of reliable methods to build confidence sets for the Aumann mean of a random closed set estimated through the Minkowski empirical mean. In order to do so, we introduce a procedure to build a confidence set based on Weil's result for the Hausdorff distance between the empirical and the Aumann means; then, we introduce another procedure based on the support function.

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“…A well known result (see Artstein and Vitale, 1975) states that RACS satisfy a Law of Large Numbers.…”

Section: Some Results On Random Setsmentioning

confidence: 99%

Confidence Sets for the Aumann Mean of a Random Closed Set

Seri

Choirat

2004

Computational Science and Its Applications – ICCSA 2004

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“…Following Artstein and Vitale [4] in adapting the Aumann [16] integral, the expectation of X is defined by { } is a selection of and .…”

Section: Notations and Preliminariesmentioning

confidence: 99%

“…The support function is one of the most important concepts in convex geometry. The goal of the present paper is to discuss a new approach for the relationship between selection expectation and support function, which has played an essential role in proving the strong law of large numbers for random compact sets [4].…”

Section: Introductionmentioning

confidence: 99%

A Note on the Selection Expectation and Support Function

He¹

2015

APM

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“…And the limit theory of set-valued random variables has been developed quite extensively. In 1975, Artstein and Vitale used an embedding theorem to prove a strong law of large numbers for independent and identically distributed set-valued random variables whose basic space is a d-dimensional Euclidean space R d (Artstein, 1975), and Hiai extended it to the case that basic space is a separable Banach space X (Hiai, 1984). Taylor and Inoue proved SLLN's for only independent case in Banach space (Taylor & Inoue, 1985).…”

Section: Introductionmentioning

confidence: 99%