De Finetti's theorem and related results for infinite weighted exchangeable sequences
Abstract
De Finetti's theorem, also called the de Finetti-Hewitt-Savage theorem, is a foundational result in probability and statistics. Roughly, it says that an infinite sequence of exchangeable random variables can always be written as a mixture of independent and identically distributed (i.i.d.) sequences of random variables. In this paper, we consider a weighted generalization of exchangeability that allows for weight functions to modify the individual distributions of the random variables along the sequence, provided that -- modulo these weight functions -- there is still some common exchangeable base measure. We study conditions under which a de Finetti-type representation exists for weighted exchangeable sequences, as a mixture of distributions which satisfy a weighted form of the i.i.d. property. Our approach establishes a nested family of conditions that lead to weighted extensions of other well-known related results as well, in particular, extensions of the zero-one law and the law of large numbers.
Keywords
Cite
@article{arxiv.2304.03927,
title = {De Finetti's theorem and related results for infinite weighted exchangeable sequences},
author = {Rina Foygel Barber and Emmanuel J. Candes and Aaditya Ramdas and Ryan J. Tibshirani},
journal= {arXiv preprint arXiv:2304.03927},
year = {2023}
}