Large deviations for sums of i.i.d. random compact sets

Abstract

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is the analog of Cramer theorem for random compact sets. Several works have been devoted to deriving limit theorems for random sets. For i.i.d. random compact sets in R, the law of large numbers was initially proved by Artstein and Vitale [1] and the central limit theorem by Cressie [3], Lyashenko [10] and Weil [16]. For generalizations to non compact sets, see also Hess [8]. These limit theorems were generalized to the case of random compact sets in a Banach space by Gine, Hahn and Zinn [7] and Puri and Ralescu [11]. Our aim is to prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, to prove the analog of the Cramer theorem. We consider a separable Banach space F with norm || ||. We denote by K(F ) the collection of all non empty compact subsets of F . For an element A of K(F ), we denote by coA the closed convex hull of A. Mazur’s theorem [5, p 416] implies that, for A in K(F ), coA belongs to coK(F ), the collection of the non empty compact convex subsets of F . The space K(F ) is equipped with the Minkowski addition and the scalar multiplication: for A1, A2 in K(F ) and λ a real number, A1 +A2 = { a1 + a2 : a1 ∈ A1, a2 ∈ A2 } , λA1 = {λa1 : a1 ∈ A1 } . 1991 Mathematics Subject Classification. 60D05, 60F10.