Necessary and sufficient conditions for convergence of stochastic approximation algorithms under arbitrary disturbances

A parameterised extension of the Kushner-Clark condition (1978) is introduced for the study of convergence of stochastic approximation algorithms. Our results provide necessary and sufficient conditions for convergence that hold in a Hilbert space setting and apply to general gain sequences. These results exhibit the interplay among the noise sequence, the gain sequence, and key properties of the underlying function. The proof is direct, completely deterministic, and is elementary, involving only basic notions of convergence. Some corollaries to our main result are also presented.