Processus al\'eatoires et applications -- Algorithmes MCMC et vitesse de convergence

TL;DR

This paper provides an in-depth overview of Markov chains, spectral theory, and MCMC algorithms, focusing on convergence speed, suitable for advanced applied mathematics students and researchers.

Contribution

It offers a comprehensive French-language lecture note covering theoretical foundations and practical algorithms for Markov chains and MCMC methods.

Findings

Analysis of spectral properties and convergence rates

Discussion of Lyapunov functions in convergence analysis

Overview of MCMC algorithms and their convergence behavior

Abstract

Lecture notes (in French) of a master 2 level course in applied mathematics. Contents: Part I. Markov chains on a countable space. 1. Examples 2. Summary of basic properties. 3. Spectral theory and speed of convergence. 4. Lyapunov functions and speed of convergence. 5. MCMC algorithms. Part II. Markov chains on continuous space. 6. Definitions and examples. 7. Invariant probability and speed of convergence.