  Given an ODE and its perturbation, the Alekseev formula expresses the
solutions of the latter in terms related to the former. By exploiting this
formula and a new concentration inequality for martingale-differences, we
develop a novel approach for analyzing nonlinear Stochastic Approximation (SA).
This approach is useful for studying a SA's behaviour close to a Locally
Asymptotically Stable Equilibrium (LASE) of its limiting ODE; this LASE need
not be the limiting ODE's only attractor. As an application, we obtain a new
concentration bound for nonlinear SA. That is, given $\epsilon >0$ and that the
current iterate is in a neighbourhood of a LASE, we provide an estimate for i.)
the time required to hit the $\epsilon-$ball of this LASE, and ii.) the
probability that after this time the iterates are indeed within this
$\epsilon-$ball and stay there thereafter. The latter estimate can also be
viewed as the `lock-in' probability. Compared to related results, our
concentration bound is tighter and holds under significantly weaker
assumptions. In particular, our bound applies even when the stepsizes are not
square-summable. Despite the weaker hypothesis, we show that the celebrated
Kushner-Clark lemma continues to hold. %
