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A nonstandard proof of de Finetti's theorem

Irfan Alam

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.

A nonstandard proof of de Finetti's theorem

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.

Paper Structure

This paper contains 1 section, 18 theorems, 52 equations.

Table of Contents

  1. Background from nonstandard analysis

Key Result

Theorem 2

Let $X_1, X_2, \ldots$ be a sequence of exchangeable Bernoulli random variables. There exists a unique measure $\mu$ on the interval $[0,1]$ such that the following holds: for any $k \in \mathbb{N}$ and $e_1, \ldots, e_k \in \{0,1\}$.

Theorems & Definitions (31)

  • Definition 1
  • Theorem 2: de Finetti
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 21 more