TABLE OF CONTENTS This chapter is dedicated to the asymptotic analysis (convergence and convergence rate) of recursive stochastic algorithms, which are used to solve problems arising in many areas (engineering, economic, biology, ecology, medicine, etc.) [1, 2, 5, 7, 9, 10, 15, 42, 44, 50, 63, 64, 67, 81]. For example, there is now a huge volume of studies dealing with the impact of maintenance activities on production, environmental protection and the reduction of accidents (consider, for example, the explosion of the chemical plant AZF on September 21, 2001). Indeed, many reliability problems lead to an optimization problem. There is no ready made machinery that we can put our algorithm into and that would produce for us the asymptotic properties by turning the crank. Nevertheless, we shall try to derive some ideas of how to state the convergence of a given algorithm, and to estimate its convergence rate. We will derive the lines of a global methodology in order to achieve this analysis. This methodology will be presented in the form of "if-then-else."
This chapter is organized as follows. The next section deals with the methodology cited above. In Section 4.3, we will present several applications of the standard inequalities (Cauchy, Jensen, Minkowski, Hadamard, inequalities based on vectors, matrices and determinants, etc.), well-known lemmas (Borel-Cantelli, Kronecker, Toeplitz, etc.) and theorems (Robbins-Monro, Robbins-Siegmund, etc.). These applications will be extracted from the proofs of different results from the literature. We will treat in detail two cases in Sections 4.4 and 4.5. The first...
