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 anan is replaced by a random matrix AnAn: Xn+1=Xn-An∇g(Xn)-AnVn.Xn+1=Xn-An∇g(Xn)-AnVn. We give two theorems of almost sure convergence in the case where the equation ∇g=0∇g=0 has a set of solutions.