1
Introduction
2
Foundations
▶
2.1
Core Definitions
2.2
\(\sigma \)-algebra Infrastructure
3
Easy Directions
▶
3.1
Exchangeable implies Contractable
3.2
Conditionally i.i.d. implies Exchangeable
4
Main Implication: Contractable implies Conditionally i.i.d.
▶
4.1
Via Martingale (Aldous’ proof)
▶
4.1.1
Pair Law Equality
4.1.2
Kallenberg Chain and Convergence
4.1.3
Factorization and Directing Measure
4.2
Via L\(^2\) (Elementary proof)
▶
4.2.1
Block Averages and Covariance Structure
4.2.2
Cesaro Convergence
4.2.3
Directing Measure Construction
4.3
Via Koopman (Mean Ergodic Theorem)
▶
4.3.1
Block Averages and Ergodic Theory
4.3.2
Contractability and Factorization
5
Common Ending
6
Main Theorem
Dependency graph
Exchangeability and de Finetti’s theorem
Cameron Freer
1
Introduction
2
Foundations
2.1
Core Definitions
2.2
\(\sigma \)-algebra Infrastructure
3
Easy Directions
3.1
Exchangeable implies Contractable
3.2
Conditionally i.i.d. implies Exchangeable
4
Main Implication: Contractable implies Conditionally i.i.d.
4.1
Via Martingale (Aldous’ proof)
4.1.1
Pair Law Equality
4.1.2
Kallenberg Chain and Convergence
4.1.3
Factorization and Directing Measure
4.2
Via L\(^2\) (Elementary proof)
4.2.1
Block Averages and Covariance Structure
4.2.2
Cesaro Convergence
4.2.3
Directing Measure Construction
4.3
Via Koopman (Mean Ergodic Theorem)
4.3.1
Block Averages and Ergodic Theory
4.3.2
Contractability and Factorization
5
Common Ending
6
Main Theorem