Limit theorems for stochastic approximation algorithms

We prove a central limit theorem applicable to one dimensional stochastic approximation algorithms that converge to a point where the error terms of the algorithm do not vanish. We show how this applies to a certain class of these algorithms that in particular covers a generalized P\'olya urn model, which is also discussed. In addition, we show how to scale these algorithms in some cases where we cannot determine the limiting distribution but expect it to be non-normal.