Large deviations for sums of i.i.d. random compact sets
Large deviations for sums of i.i.d. random compact sets
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DOI：
10.1090/s0002-9939-99-04788-7
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发表时间：
1999-04
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通讯作者：
R. Cerf
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作者：
R. Cerf
我们证明了 i.i.d 的 Minkowski 和的大偏差原理。 Banach 空间中的随机紧集，类似于随机紧集的 Cramer 定理。有几部著作致力于推导随机集的极限定理。对于 i.i.d. R 中的随机紧集，大数定律最初由 Artstein 和 Vitale [1] 证明，中心极限定理由 Cressie [3]、Lyashenko [10] 和 Weil [16] 最初证明。对于非紧集的推广，另请参见 Hess [8]。 Gine、Hahn 和 Zinn [7] 以及 Puri 和 Ralescu [11] 将这些极限定理推广到 Banach 空间中的随机紧集情况。我们的目标是证明 i.i.d 的 Minkowski 和的大偏差原理。 Banach空间中的随机紧集，即证明Cramer定理的类比。我们考虑一个可分离的 Banach 空间 F，范数 || ||。我们用 K(F ) 表示 F 的所有非空紧子集的集合。对于 K(F ) 的元素 A，我们用 coA 表示 A 的闭凸包。Mazur 定理 [5, p 416] 意味着，对于 K(F ) 中的 A，coA 属于 coK(F )，即 F 的非空紧凸子集的集合。空间 K(F ) 配备了闵可夫斯基加法和标量乘法：对于 K(F ) 中的 A1、A2 和 λ 实数，A1 +A2 = { a1 + a2 : a1 ∈ A1, a2 ∈ A2 } ， λA1 = {λa1 : a1 ∈ A1 } 。 1991年数学学科分类。 60D05、60F10。
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.