## Abstract

A length-n random sequence X_{1},..., X_{n} in a space S is finitely exchangeable if its distribution is invariant under all n! permutations of coordinates. Given N >n, we study the extendibility problem: When is it the case that there is a length-N exchangeable random sequence Y_{1},...,Y_{N} so that (Y_{1},..., Y_{n}) has the same distribution as (X_{1},..., X_{n})? In this paper, we give a necessary and sufficient condition so that, for given n and N, the extendibility problem admits a solution. This is done by employing functional-analytic and measure-theoretic arguments that take into account the symmetry. We also address the problem of infinite extendibility. Our results are valid when X_{1} has a regular distribution in a locally compact Hausdorff space S. We also revisit the problem of representation of the distribution of a finitely exchangeable sequence.

Original language | English |
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Pages (from-to) | 7067-7092 |

Number of pages | 26 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

- Bounded linear functional
- De Finetti
- Exchangeable
- Extendible
- Finitely exchangeable
- Hahn–Banach
- Permutation
- Set function
- Signed measure
- Symmetric
- U-statistics
- Urn measure