Large Deviations for Sums of I.i.d. Random Compact Sets

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 Cramér 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 noncompact sets, see also Hess [8]. These limit theorems were generalized to the case of random compact sets in a Banach space by Giné, 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 Cramér theorem. We consider a separable Banach space F with norm || ||. We denote by K(F ) the collection of all nonempty 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 ), co A belongs to coK(F ), the collection of the nonempty 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 } . The Hausdorff distance d(A1, A2) = max { sup a1∈A1 inf a2∈A2 ||a1 − a2||, sup a2∈A2 inf a1∈A1 ||a2 − a1|| } makes (K(F ), d) a complete separable metric space (i.e., a Polish space). We endow K(F ) with the Borel σ–field associated to the Hausdorff topology. We denote by F ∗ the topological dual of F and by B∗ the unit ball of F ∗. The Banach–Alaoglu theorem asserts that B∗ endowed with the weak∗ topology w∗ is compact [13]. Moreover the space (B∗, w∗) is separable and metrizable. We denote by M(B∗) the set of Borel signed measures on B∗ (the σ–field being the σ–field generated by the weak∗ topology). Let (Ω,F , P ) be a probability space. A random compact set of F is a measurable function from Ω to K(F ), i.e., a random variable with values in K(F ). Received by the editors September 10, 1997 and, in revised form, October 27, 1997. 1991 Mathematics Subject Classification. Primary 60D05, 60F10.