RATE OF CONVERGENCE FOR COMPUTING EXPECTATIONS OF STOPPING FUNCTIONALS OF AN ALPHA-MIXING PROCESS

The shift method consists in computing the expectation of an integrabl e functional F defined on the probability space ((R-d)(N), B(R-d)(xN), mu(xN)) (mu is a probability measure on R-d) using Birkhoff's Pointwi se Ergodic Theorem, i.e.1/n (k=0)Sigma(n-1)Fo theta(k) --> E(F) a.s. a s n --> +infinity, where theta denotes the canonical shift operator. W hen F lies in L-2(F-T, mu(xN)) for some integrable enough stopping tim e T, several weak (CLT) or strong (Gal-Koksma Theorem or LIL) convergi ng rates hold. The method successfully competes with Monte Carlo. The aim of this paper is to extend these results to more general probabili ty distributions P on ((R-d)(N), B(R-d)(xN)), namely when the canonica l process (X-n)(n is an element of N) is P-stationary, alpha-mixing an d fulfils Ibragimov's assumption n greater than or equal to 0 Sigma al pha(delta/(2+delta))(n)<+infinity for some delta > 0. One application is the computation of the expectation of functionals of an alpha-mixin g Markov Chain, under its stationary distribution P-nu. It may both pr ovide a better accuracy and save the random number generator compared to the usual Monte Carlo or shift methods on independent innovations.