The almost equivalence of pairwise and mutual independence and the duality with exchangeability
The first aim of this paper is to show that all the notions of independence are, in fact, almost identical to their pairwise counterpart in an ideal setting. An asymptotic interpretation shows that in the usual sequential setting, these notions are asymptotically equivalent even though they are still different. As a consequence of this type of equivalence result, it is possible to unify multiple versions of various, seemingly unrelated, multiplicative properties of random variables, such as those involving generating functions, characteristic functions, and maximum of random variables. The second aim is to show that the notions of independence and exchangeability are dual to each other in the sense that almost mutual independence (almost exchangeability) of the random variables in a process in an ideal setting is equivalent to almost exchangeability (almost mutual independence) of the sample functions of the process. Such a duality result can also be interpreted in the asymptotic setting by routine techniques. As another instance of the phenomenon ``Two implies many'', pairwise and multiple versions of exchangeability are also shown to be almost equivalent. 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.
scientific article; zbMATH DE number 1233917