Almost sure and Lq-convergence of the re-initialized BMP scheme

We consider stochastic approximation algorithms with Markovian dynamics introduced by Benveniste, Metivier and Priouret (BMP). A major deficiency of the BMP theory is that it guarantees convergence only with probability strictly less than 1. This deficiency will be overcome by incorporating a resetting mechanism for the parameter with a fairly arbitrary truncation domain. At the same time the state is also reset. The algorithm is shown to converge to the assumed unique stationary point of the associated ODE with probability 1. The result is complementary to earlier results using resetting. An outline of the basic technical aspects of the BMP theory will be also given. Finally, the basic ideas for establishing L_q-convergence of the estimation error, including rate of convergence, will be presented.