Mathematical remark: the rigorous demonstration of the expression of the density of probability formula_17 is not as direct as it appears above: from the application of the usual (1D) central limit theorem one can deduce that formula_18, formula_19 and formula_9 are distributed according to a centered normal distribution of variance formula_21. Then, the expression given above for formula_17 is not the only one that is compatible with such distribution for formula_18, formula_19 and formula_9. However, since the components of the vectors formula_3 are uncorrelated for the random walk we are considering, it follows that formula_18, formula_19 and formula_9 are also uncorrelated. This additional condition can only be fulfilled if formula_2 is distributed according to formula_17. Alternatively, this result can also be demonstrated by applying a multidimensional generalization of the central limit theorem, or through symmetry arguments.