On the convergence of linear stochastic approximation procedures

Many stochastic approximation procedures result in a stochastic algorithm of the form h_k+1=h_k+1/k(b_k-A _kh_k), for all k=1,2,3,... . Here, {b_k,k=1,2,3...} is a R^d-valued process, {A_k,k=1,2,3,...} is a symmetric, positive semidefinite R e^d×d-valued process, and {h_k,k=1,2,3,...} is a sequence of stochastic estimates which hopefully converges to h^Δ=[lim_N→∞/1 NΣ_k=1^NEA_k]^-1 {lim_N→∞/1NΣ_k=1^NEb _k} (assuming everything here is well defined). We give an elementary proof which relates the almost sure convergence of {h_k ,k=1,2,3,...} to strong laws of large numbers for {b_k,k=1,2,3,...} and {A_k,k=1,2,3,...}