On the convergence of markovian stochastic algorithms with rapidly decreasing ergodicity rates

We analyse the convergence of stochastic algorithms with Markovian noise when the ergodicity of the Markov chain governing the noise rapidly decreases as the control parameter tends to infinity. In such a case, there may be a positive probability of divergence of the algorithm in the classic Robbins-Monro form. We provide sufficient condition which ensure convergence. Moreover, we analyse the asymptotic behaviour of these algorithms and state a diffusion approximation theorem