Convergence of stochastic approximations with state dependent noise under weak conditions

A new method is presented for quickly getting the ODE (ordinary differential equation) associated with the asymptotic properties of the stochastic approximation Xn+1 = Xn + an f(Xn, ??n) (or the projected algorithm). The method requires that { Xn, ??n-1} be Markov with a "Feller" transition function, but little else, except that if Xn ?? x, the process {??n(x), n ?? 0} have a unique invariant measure (and even the uniqueness can be weakened). No mixing condition is required, nor the construction of averaged test functions, and f(??,??) need not be continuous. A detailed analysis of the way that {??n} varies with {Xn} is not required. For the class of sequences treated, the conditions seem easier to verify than for other methods. An example illustrates the power of the approach.