  Among Markovian processes, the hallmark of L\'evy flights is superdiffusion,
or faster-than-Brownian dynamics. Here we show that L\'evy laws, as well as
Gaussians, can also be the limit distributions of processes with long range
memory that exhibit very slow diffusion, logarithmic in time. These processes
are path-dependent and anomalous motion emerges from frequent relocations to
already visited sites. We show how the Central Limit Theorem is modified in
this context, keeping the usual distinction between analytic and non-analytic
characteristic functions. A fluctuation-dissipation relation is also derived.
Our results may have important applications in the study of animal and human
displacements.
