Stochastic approximation and the final value theorem

The aim here is to show how to obtain many of the well-known limit results (i.e., central limit theorem, law of the iterated logarithm, invariance principle) of stochastic approximation (SA) by a shorter argument and under weaker conditions. The idea is to introduce an artificial sequence, related to the SA scheme, and which clearly obeys the limit law. This sequence is subtracted from the SA scheme and then simple deterministic limit theory is used to show the remainder is negligible. As a consequence of this approach proofs are shorter and the meaning of conditions becomes clearer. Because the difference equations are not summed up it is simple to state results for general a_n, c_n sequences.