  We study a scenario under which variable step random walks give anomalous
statistics. We begin by analyzing the Martingale Central Limit Theorem to find
a sufficient condition for the limit distribution to be non-Gaussian. We note
that the theorem implies that the scaling index $\zeta$ is 1/2. For
corresponding continuous time processes, it is shown that the probability
density function $W(x;t)$ satisfies the Fokker-Planck equation. Possible forms
for the diffusion coefficient are given, and related to $W(x,t)$. Finally, we
show how a time-series can be used to distinguish between these variable
diffusion processes and L\'evy dynamics.
