Abstract. Random walk simulation of the Levy flight shows a linear relation between the mean square displacement and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value (l m ) affects the diffusion coefficient, even though it remains constant for l m 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 . 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.

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A new analytical solution is developed for the one-and two-dimensional generalized discrete time random walk with discrete and finite length of jumping. It is shown that the ensemble average of the mean square displacement has a linear relation with time. This result is general and independent of mathematical form of distribution function but magnitude of diffusion coefficient depends on the distribution function. This result confirms our previous numerical work. The truncated Levy flight is studied as a special case and the diffusion coefficient for different maximum length of jump is calculated numerically. The obtained results are in good agreement with the analytical solution.