Consistency of stochastic approximation algorithm with quasi-associated random errors

Many mathematical and physical problems are led to find a root of a real function f. This kind of equation is an inverse problem and it is difficult to solve it. Especially in engineering sciences, the analytical expression of the function f is unknown to the experimenter, but it can be measured at each point x(k) with M(x(k)) as expected value and induced error (k). The aim is to approximate the unique root under some assumptions on the function f and errors (k). We use a stochastic approximation algorithm that constructs a sequence (x(k))(k 1). We establish the almost complete convergence of the sequence (x(k))(k) to the exact root by considering the errors ((k))(k) quasi-associated and we illustrate the method by numerical examples to show its efficiency.