  This paper takes the so-called probabilistic approach to the Strong Renewal
Theorem (SRT) for multivariate distributions in the domain of attraction of a
stable law. A version of the SRT is obtained that allows any kind of
lattice-nonlattice composition of a distribution. A general bound is derived to
control the so-called "small-$n$ contribution", which arises from random walk
paths that have a relatively small number of steps but make large cumulative
moves. The asymptotic negligibility of the small-$n$ contribution is essential
to the SRT. Applications of the SRT are given, including some that provide a
unified treatment to known results but with substantially weaker assumptions.
