Stability and instability of limit points for stochastic approximation algorithms

It is shown that the limit points of a stochastic approximation (SA) algorithm consist of a connected set. Conditions are given to guarantee the uniqueness of the limit point for a given initial value. Examples are provided wherein {x_n} of SA algorithm converges to a limit x¯ independent of initial values, but x¯ is unstable for the differential equation x˙=f(x) with a nonnegative Lyapunov function. Finally, sufficient conditions are given for stability of x˙=f(x) at x¯ if {x_n} tends to x¯ for any initial values