Approach for a metric space with a convex combination operation and applications

Abstract

Abstract. In this paper, we embed metric space endowed with a convex combination operation, named convex combination space, into a Banach space and the embedding preserves the structures of metric and convex combination. For random element taking values in this kind of space, applications of embedding are also established. On the one hand, some nice properties of expectation such as representation of expected value through continuous affine mappings, the linearity of expectation will be given. On the other hand, the notion of conditional expectation will be also introduced and discussed. Thanks to embedding theorem, we establish some basic properties of conditional expectation, Jensen's inequality, convergences of martingales and ergodic theorem.Mathematics Subject Classifications (2000): 60B05, 60F15, 51K05.

Minimal faces and Schur's Lemma for embeddings into $R^\mathcal{U}$

“…In [2], N. Brown exhibits a convex structure on the space of unitary equivalence classes of embeddings of a II 1 -factor in an ultrapower of the separable hyperfinite II 1 -factor. Since its appearance in 2011, the convex structure in [2] has received a fair amount of attention in the literature-see [4], [3], [6], [7], [8], [9], [5], [10], and [1]. Part of the appeal of Brown's work is that it links convex geometric concepts with operator algebraic ones.…”

Baum-Katz’s Type Theorems for Pairwise Independent Random Elements in Certain Metric Spaces

“…Since then, some limit theorems for random elements taking values in convex combination space were considered and extended (see [9,11,12,14]). On the other hand, as shown recently in [13], it is fairly remarkable that although these spaces are not linear in general, they always contains a subspace which can be isometrically embedded into a Banach space and this embedding preserves the convex combination operation.…”

“…Based on this property, if a result holds in CC space, then it can be uplifted to the space of nonempty compact subsets. Further details can be found in [11,12,13]. The notion of compactly uniform integrability in Cesàro sense for a collection of random elements taking values in Banach spaces was discussed by many authors (see, e.g., [1,3,15]).…”

“…(b) Notice that some arguments in the proof of Theorem 3.1 can be also obtained by combining the embedding theorem [13,Theorem 3.3] and corresponding results in Banach space. However, this embedding is not too straightforward since its proof requires several intermediate results, while the proof of Theorem 3.1 presented above is more direct.…”