De Finetti's theorem and related results for infinite weighted exchangeable sequences

TL;DR

This paper extends De Finetti's theorem to weighted exchangeable sequences, establishing conditions for representing such sequences as mixtures of weighted i.i.d. distributions, and generalizes related probabilistic laws.

Contribution

It introduces a weighted generalization of exchangeability and provides conditions for a De Finetti-type representation in this new setting.

Findings

Weighted exchangeability can be characterized by a common base measure with weights.

Conditions are established for the existence of a weighted De Finetti representation.

Extensions of zero-one law and law of large numbers are derived for weighted 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.