We prove convergence with probability one of a multivariate Markov stochastic approximation procedure of the Robbins-Monro type with several roots. The argument exploits convergence of the corresponding system of ordinary differential equations to its stationary points. If the points are either linearly stable or linearly unstable, we prove convergence with probability 1 of the procedure to a random vector whose distribution concentrates on the set of stable stationary points. This generalizes for procedures with several roots the approach suggested by L. Ljung for processes with a single root.
Along with stochastic approximation processes as such, the result can be applied to generalized urn schemes and stochastic models of technological and economic dynamics based on them, in particular, evolutionary games with incomplete information
