The almost equivalence of pairwise and mutual independence and the duality with exchangeability

• 成果类型: Article
• 署名作者: Sun, YE
• 署名单位: National University of Singapore; Yale University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050196
• 发表日期: 1998
• 页码: 425-456
• 关键词: nonstandard

摘要

For a large collection of random variables in an ideal setting, pairwise independence is shown to be almost equivalent to mutual independence. An asymptotic interpretation of this fact shows the equivalence of asymptotic pairwise independence and asymptotic mutual independence for a triangular array (or a sequence) of random variables. Similar equivalence is also presented for uncorrelatedness and orthogonality as well as for the constancy of joint moment functions and exchangeability. General unification of multiplicative properties for random variables are obtained. The duality between independence and exchangeability is established through the random variables and sample functions in a process. Implications in other areas are also discussed, which include a justification for the use of mutually independent random variables derived from sequential draws where the underlying population only satisfies a version of weak dependence. Macroscopic stability of some mass phenomena in economics is also characterized via almost mutual independence. It is also pointed out that the unit interval can be used to index random variables in the ideal setting, provided that it is endowed together with some sample space a suitable larger measure structure.