Rate of convergence for computing expectations of stopping functionals of an α-mixing process

Abstract

The shift method consists in computing the expectation of an integrable functional F defined on the probability space ((ℝ d ) N , B(ℝ d )⊗N , μ⊗N ) (μ is a probability measure on ℝ d ) using Birkhoff's Pointwise Ergodic Theorem, i.e. as n → ∞, where θ denotes the canonical shift operator. When F lies in L 2( F T , μ⊗N ) for some integrable enough stopping time T, several weak (CLT) or strong (Gàl-Koksma Theorem or LIL) converging rates hold. The method successfully competes with Monte Carlo. The aim of this paper is to extend these results to more general probability distributions P on ((ℝ d ) N , B(ℝ d )⊗N ), namely when the canonical process (X n ) n∊N is P-stationary, α-mixing and fulfils Ibragimov's assumption for some δ > 0. One application is the computation of the expectation of functionals of an α-mixing Markov Chain, under its stationary distribution P ν. It may both provide a better accuracy and save the random number generator compared to the usual Monte Carlo or shift methods on independent innovations.