Summary
Call a sequence {X n } of r.v.'s ε-exchangeable if on the same probability space there exists an exchangeable sequence {Y n } such thatP(|X n −Y n |≧ε)≦ε for alln. We prove that any tight sequence {X n } defined on a rich enough probability space contains ε-exchangeable subsequences for every ε>0. The distribution of the approximating exchangeable sequences is also described in terms of {X n }. Our results give a convenient way to prove limit theorems for subsequences of general r.v. sequences. In particular, they provide a simplified way to prove the subsequence theorems of Aldous [1] and lead also to various extensions.
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Berkes, I., Péter, E. Exchangeable random variables and the subsequence principle. Probab. Th. Rel. Fields 73, 395–413 (1986). https://doi.org/10.1007/BF00776240
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DOI: https://doi.org/10.1007/BF00776240