Weak convergence for stochastic approximation algorithms with bounded but discontinuous driving forces

Stochastic approximation algorithms of the form xn+1 = xn + an G(xn, En) are treated where G(x, E) is bounded but might be discontinuous. The method of averaging and Liapunov's direct method are combined to get tightness of {xn, n ≥ 0} and weak convergence result of xn, as n goes to +∞.