Rate of convergence of iterative algorithms

Abstract

In the previous chapter, we showed how to simulate a distribution as the limit of an iterative algorithm. In practice, these algorithms cannot be run eternally; they have to be stopped after a finite number of iterations. Of course there is no reason for the distribution thus simulated to be identical to the limit distribution, and the number of iterations to carry out must be chosen so that their difference lies below a prescribed level of acceptability. This requires the study of the rate of convergence of the iterative algorithm. A considerable amount of literature has been devoted to the determination of the rate of convergence of algorithms based on Markovian iterations (in particular Nummelin, 1984; Meyn and Tweedie, 1993; Tierney, 1994; Duflo, 1996). This chapter only deals with two cases corresponding to two different assumptions on the transition kernel of the algorithm:

• If the transition kernel is minorized by a positive measure, then the rate of convergence is uniform (section 9.2).

• If the transition kernel admits an isofaetorial representation, then a geometric rate of convergence is expected in many cases (section 9.3). The integral range introduced in chapter 4 can be used to determine it empirically (section 9.4).