Almost sure convergence of stochastic gradient processes with matrix step sizes

We consider a stochastic gradient process, which is a special case of stochastic approximation process, where the positive real step size an is replaced by a random matrix An: Xn+1=Xn-An[backward difference]g(Xn)-AnVn. We give two theorems of almost sure convergence in the case where the equation [backward difference]g=0 has a set of solutions.