  Commonly, normal diffusive behavior is characterized by a linear dependence
of the second central moment on time, $< x^2(t) >\propto t$, while anomalous
behavior is expected to show a different time dependence, $ < x^2(t) > \propto
t^{\delta}$ with $\delta <1$ for subdiffusive and $\delta >1$ for
superdiffusive motions. Here we demonstrate that this kind of qualification, if
applied straightforwardly, may be misleading: There are anomalous transport
motions revealing perfectly "normal" diffusive character ($< x^2(t) >\propto
t$), yet being non-Markov and non-Gaussian in nature. We use recently developed
framework \cite[Phys. Rev. E \textbf{75}, 056702 (2007)]{magdziarz2007b} of
Monte Carlo simulations which incorporates anomalous diffusion statistics in
time and space and creates trajectories of such an extended random walk. For
special choice of stability indices describing statistics of waiting times and
jump lengths, the ensemble analysis of paradoxical diffusion is shown to hide
temporal memory effects which can be properly detected only by examination of
formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov
equation).
