A nonstandard proof of de Finetti's theorem

TL;DR

This paper presents a novel nonstandard analytic proof of de Finetti's theorem for exchangeable Bernoulli sequences, demonstrating the representation as a mixture of iid Bernoulli sequences using hyperfinite sample means.

Contribution

It introduces a nonstandard analytic approach to prove de Finetti's theorem, offering a new perspective beyond traditional proofs.

Findings

The proof confirms the representation of exchangeable Bernoulli sequences as mixtures of iid sequences.

Uses combinatorial arguments to relate probability distributions to hyperfinite sample means.

Provides an alternative proof method leveraging nonstandard analysis.

Abstract

We give a nonstandard analytic proof of de Finetti's theorem for an exchangeable sequence of Bernoulli random variables. The theorem postulates that such a sequence is uniquely representable as a mixture of iid sequences of Bernoulli random variables. We use combinatorial arguments to show that this probability distribution is induced by a hyperfinite sample mean.