Martingale transforms with non-atomic limits and stochastic approximation

Summary A sufficient condition for the limit of a martingale transform to posses a continuous distribution is given. The result is used to show that for a stochastic approximation procedure, if the adjustment rate is too small then it would not converge to the target value a.s. Furthermore, if the adjustment rate is taken to be 1/n as usual but the derivative of the regression function at the target value is 0, then the convergence rate is shown to be logn instead of, the rate obtained when the derivative is non-zero.Research supported by the National Science Council, R.O.C.