  Many physical and biological processes are modeled by "particles" undergoing
L\'evy random walks. A feature of significant interest in these systems is the
mean square displacement (MSD) of the particles. Long-time asymptotic
approximations of the MSD have been established, via the Tauberian Theorem, for
systems in which the distribution of the step durations is asymptotically a
power law of infinite variance. We extend these results, using elementary
analysis, and obtain closed-form expressions as well as power law bounds for
the MSD in equilibrium, and representations of the MSD as sums of super-linear,
linear, and sub-linear terms. We show that the super-linear components are
determined by the mean and asymptotics of the step durations, but that the
linear and sub-linear components (whose size has implications for the accuracy
of the asymptotic approximation) depend on the entire distribution function.
