  We study the distribution of the maximum $M$ of a random walk whose
increments have a distribution with negative mean and belonging, for some
$\gamma>0$, to a subclass of the class $\mathcal{S}_\gamma$--see, for example,
Chover, Ney, and Wainger (1973). For this subclass we give a probabilistic
derivation of the asymptotic tail distribution of $M$, and show that extreme
values of $M$ are in general attained through some single large increment in
the random walk near the beginning of its trajectory. We also give some results
concerning the ``spatially local'' asymptotics of the distribution of $M$, the
maximum of the stopped random walk for various stopping times, and various
bounds.
