Analytical representations of non-Gaussian laws of random walks

An analytical representation of a random process with independent increments in some space (random walks introduced by Pearson) is considered. The law of random walk distribution in space is derived from the general representation of stochastic elementary hops (distribution law of hop probability) using Kadanoff’s concept of the unit increment as one hop. For limited hop laws and laws of hop distributions with all moments there naturally arises Chandrasekhar’s result that describes ordinary physical diffusion. For laws of hop distributions without the second and highest moments there also arise known Lévy walks (flights) sometimes treated as superdiffusion. For the intermediate case, where the distributions of hops have at least the second moment and not all finite moments (these hops are sometimes called truncated Lévy walks), the asymptotic form of the random walk distribution was obtained for the first time. The results obtained are compared with the experimental laws known in econophysics. Satisfactory agreement is observed between the developed theory and the empirical data for insufficiently studied truncated Lévy walks.