M2RI: Convergence of probability measures, limit theorems and optimal transport.

Description:

Objectives :

• To discover the topology of weak convergence on Polish spaces.
• To get used to limit theorems in infinite dimensions.
• To complete your understanding of probability measures and their geometry thanks to optimal transport.

Key words : Convergence in law, Polish spaces, Prokhorov's theorem, Donsker's theorem, Optimal transport.

Content :

Part 1: Convergence of probability measures
• Probability measures on Polish spaces
• Convergence in law / Weak topology
• Prokhorov's theorem

Part 2: Limit theorems in infinite dimensions
• Weak compactness in the space of continuous functions
• Donsker's theorem

Part 3: Optimal transport
• Kantorovich duality
• Brenier's theorem
• Sinkhorn algorithm

References:

• P. Billingsley. Convergence of probability measures.
• C. Villani. Topics in optimal transport.

Exam :

• Midterm: None
• Final : Written exam (3h)

Material: [ Link ]