The closeness of the range of a probability on a certain system of random events -- an elementary proof

An elementary combinatorial method is presented which can be used for proving the closeness of the range of a probability on specific systems, like the set of all linear or affine subsets of a Euclidean space. The motivation for this note came from the second author’s research in statistics: high breakdown point estimation in linear regression. By a probability distribution P , defined on the Borel σ-field of R, a collection of regression design points is represented; then, a system V of Borel subsets of R is considered. Typical examples of V are, for instance, the system V1 of all linear, or V2 of all affine proper subspaces of R. The question (of some interest in statistical theory) is: Is there an E0 ∈ V such that P (E0) = sup{P (E) : E ∈ V}? (1) For some of V , the existence of a desired E0 can be established using that (a) V is compact in an appropriate topology; (b) P is lower semicontinuous with respect to the same topology. The construction of the topology may be sometimes tedious; moreover the method does not work if, possibly, certain parts of V are omitted, making V noncompact. Also, a more general problem can be considered: Is the range {P (E) : E ∈ V} closed? (2) The positive answer to (2) implies the positive one to (1). The method outlined by (a) and (b) cannot answer (2) — we have only lower semicontinuity, not full continuity. Received by the editors November 1996. Communicated by M. Hallin. Bull. Belg. Math. Soc. 4 (1997), 621–624 622 V. Balek – I. Mizera Nevertheless, an elementary method provides the desired answer, for general P and V . The method does not require a topologization of V , and it works also for various, possibly noncompact, subsets of V . The main idea can be regarded as an extension of a simple fact that the probabilities of pairwise disjoint events cannot form a strictly increasing sequence. Linear subspaces are not disjoint; however, the intersection of two distinct ones with the same dimension is a subspace with a lower dimension. Iterating this process further, we arrive to the unique null-dimensional subspace. If, say, instead of linear subspaces the affine ones are considered, the method works in a similar way — only the terminal level is slightly different. A well-known related property — to be found, for instance, in [1], Ch. II, Ex. 48–50 — says that the range {P (E) : E ∈ S} is closed for every probability space (Ω,S, P ). However, here the background is different: probabilities of general events can form an increasing sequence — this is not true in our setting. Theorem. Let (Ω,S, P ) be a probability space. If A0 ⊆ A1 ⊆ · · · ⊆ An are sets of events such that cardA0 = 1 and for every k = 1, 2, . . . , n, the intersection of two distinct events from Ak belongs to Ak−1, then the set {P (E) : E ∈ An} is closed. Corollary. Under the assumptions of Theorem, (1) is true with V = An. Applying Theorem for V = V1, we set n = p − 1; Ak consists of all proper subspaces of dimension less or equal to k. Note that An = V1 and A0 = {0}; the other assumptions hold as well. According to Theorem, the range of P on An is closed and the supremum is attained. The cases of other V are treated in an analogous way. We shall call a system A0,A1, . . . ,An satisfying the assumptions of Theorem an intersection system. Suppose that B is a set of events such that B ⊆ An. If A0,A1, . . . ,Aν is another intersection system such that B ⊆ Aν, we can form an intersection system A′′ 0,A 1, . . . ,A′′ m by taking consecutivelyA′′ m = An∩Aν , A′′ m−1 = An−1 ∩ Aν−1, . . . , identifying A′′ 0 with the first set with cardinality 1 obtained in this process. As a result, we have m ≤ min(n, ν) and B ⊆ A′′ m. The similar construction can be carried out with more than two intersection systems; if there is any intersection system A0,A1, . . . ,An such that B ⊆ An, then the intersection of all intersection systems with this property will be called the intersection system generated by B. Note that for all k, the set Ak−1 contains exactly all pairwise intersections of events from Ak. Hence if Ak is finite, so is Ak−1. If Ak is (at most) countable, so is Ak−1. Let 1 ≤ k ≤ n. An intersection system is said to satisfy a finiteness condition at level k, if any event from Ak−1 is a subset of at most a finite number of events from Ak. Note that if the finiteness condition is satisfied at level k and Ak is infinite, so is Ak−1. As a consequence, an intersection system with infinite An cannot satisfy the finiteness condition at all levels k = 1, 2, . . . , n. Lemma. Suppose that the intersection system A0, A1, . . . , An generated by {E1, E2, . . .} satisfies the finiteness condition at levels k = 2, . . . , n and A0 = {∅}. Then limi→∞ P (Ei) = 0. Proof. By assumptions, A1,A2, . . . ,An are countably infinite. For any F ∈ Ak, k = 1, 2, . . . , n, let F = F r ⋃Ak−1. Note that F = F for F ∈ A1, since A0 = {∅}. The closeness of the range of a probability on a certain system of random events623 For all k, the elements of {F : F ∈ Ak} are pairwise disjoint. Fix e > 0. Pick B1 ⊆ A1 such that A1 r B1 is finite and