Stochastic Approximation with Discontinuous Dynamics and State Dependent Noise: w.p.1 Convergence.

Stochastic approximations might not be continuous and the noise sequence xi sub n might depend on X sub n. An averaging and an ordinary differential equation method are combined to get w.p.1 convergence for both the above algorithm and for the case where the iterates are projected back onto a bounded set G if they ever leave it. Two examples are developed, the first being an automata problem where the dynamics are not smooth and the noise is state dependent, and the second a Robbins-Monro process with observation averaging which causes the noise to be state dependent. Each example is typical of a larger class.