[论文解读] Strong Law of Large Numbers for Random Sets in Banach spaces

Vladimir Kadets, Olesia Zavarzina|arXiv (Cornell University)|Oct 7, 2024

Probability and Risk Models被引用 19

一句话总结

论文分析在巴拿赫空间的随机集序列中，哪几种强大数定律（SLLN）成立，结果表明在无限维度下全形式不成立，但在 B-凸空间中减小形式成立，在有限维度中全形式成立。

ABSTRACT

The Strong Law of Large Numbers (SLLN) for random variables or random vectors with different mathematical expectations easily reduces by means of shifts to SLLN for random variables or random vectors whose mathematical expectations are equal to zero. The situation changes for random sets, where shifts cannot reduce sets of more than one point to the set $\{0\}$. We study effects that appear because of this difference.

研究动机与目标

• Motivate how SLLN for random sets diverges from SLLN for random vectors due to nonzero expectations after shifts.
• Determine the validity of full, intermediate, and reduced SLLN forms for random convex sets in Banach spaces.
• Identify conditions under which reduced SLLN holds in B-convex spaces and finite-dimensional subspaces.

提出的方法

• Utilize the Rådström embedding to convert random sets into Banach-space valued random vectors for analysis.
• Employ combinatorial and geometric lemmas (Auerbach’s lemma, Lemma 3.3) to construct counterexamples in infinite-dimensional spaces.
• Prove full SLLN in finite-dimensional spaces via embedding and compactness arguments.
• Develop a separable reduction framework to extend reduced SLLN to non-separable spaces.
• Prove reduced SLLN in B-convex spaces by showing random sets reduce to random vectors when expectations are zero.

实验结果

研究问题

• RQ1 Does the full form of SLLN hold for random sets in infinite-dimensional Banach spaces?
• RQ2 Can the reduced form of SLLN be extended to non-separable Banach spaces via separable reduction?
• RQ3 Under what conditions does the intermediate form of SLLN hold for random sets in B-convex spaces?
• RQ4 Is the full SLLN valid for finite-dimensional random sets when their expectations lie in a finite-dimensional subspace?
• RQ5 What are the limitations and open questions for SLLN forms in random sets (non-identical distributions, non-convexity)?

主要发现

• The full form of SLLN for random sets is not valid in any infinite-dimensional Banach space, even for convex finite-dimensional values.
• The full form of SLLN holds in finite-dimensional spaces for uniformly bounded independent random sets with given expectations.
• The reduced form of SLLN is valid in every B-convex space, since zero-expectation random sets reduce to random vectors.
• In B-convex spaces, the intermediate form of SLLN holds when the common expectation lies in a finite-dimensional subspace.
• A separable reduction lemma enables extending reduced SLLN results from separable to non-separable spaces.