On array-RQMC for Markov chains: mapping alternatives and convergence rates

Pierre L'Ecuyer (1) , Christian Lécot (2, 3) , Adam L'Archevêque-Gaudet (1)

1 DIRO - Département d'Informatique et de Recherche Opérationnelle [Montreal]
2 LAMA - Laboratoire de Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc

We study the convergence behavior of a randomized quasi-Monte Carlo (RQMC) method for the simulation of discrete-time Markov chains, known as array-RQMC. The goal is to estimate the expectation of a smooth function of the sample path of the chain. The method simulates n copies of the chain in parallel, using highly uniform point sets randomized independently at each step. The copies are sorted after each step, according to some multidimensional order, for the purpose of assigning the RQMC points to the chains. In this paper, we provide some insight on why the method works, explain what would need to be done to bound its convergence rate, discuss and compare different ways of realizing the sort and assignment, and report empirical experiments on the convergence rate of the variance and of the mean square discrepancy between the empirical and theoretical distribution of the states, as a function of n , for various types of discrepancies.

Domaines: Analyse numérique [math.NA]

Pierre L'Ecuyer, Christian Lécot, Adam L'Archevêque-Gaudet. On array-RQMC for Markov chains: mapping alternatives and convergence rates. Eighth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Jul 2008, Montréal, Canada. pp.485-500. ⟨hal-00392299⟩