On equivalence of some noise conditions for stochastic approximation algorithms

We study four conditions on noise sequences for convergence of stochastic approximation algorithms on a general Hilbert space: Kushner and Clark's condition (1978), Chen's condition (1994), Kulkarni and Horn's condition (1995), and a decomposition condition. We discuss various properties of these conditions. In our main result we show that the four conditions are all equivalent, and are both necessary and sufficient for convergence of stochastic approximation algorithms under appropriate assumptions.