  It is shown that under standard hypotheses, if stochastic approximation
iterates remain tight, they converge with probability one to what their o.d.e.
limit suggests. A simple test for tightness (and therefore a.s. convergence) is
provided. Further, estimates on lock-in probability, i.e., the probability of
convergence to a specific attractor of the o.d.e. limit given that the iterates
visit its domain of attraction, and sample complexity, i.e., the number of
steps needed to be within a prescribed neighborhood of the desired limit set
with a prescribed probability, are also provided. The latter improve
significantly upon existing results in that they require a much weaker
condition on the martingale difference noise.
