  Random walk simulation of the Levy flight shows a linear relation between the
mean square displacement <r2> and time. We have analyzed different aspects of
this linearity. It is shown that the restriction of jump length to a maximum
value (lm) affects the diffusion coefficient, even though it remains constant
for lm greater than 1464. So, this factor has no effect on the linearity. In
addition, it is shown that the number of samples does not affect the results.
We have demonstrated that the relation between the mean square displacement and
time remains linear in a continuous space, while continuous variables just
reduce the diffusion coefficient. The results are also implied that the
movement of a levy flight particle is similar to the case the particle moves in
each time step with an average length of jumping <l>. Finally, it is shown that
the non-linear relation of the Levy flight will be satisfied if we use time
average instead of ensemble average. The difference between time average and
ensemble average results points that the Levy distribution may be a non-ergodic
distribution.
