Distribution of returns and its asymptotic behavior

Motivated by the concept of self-organized criticality and its link with the theory of random walks, distributions of returns are studied. In particular, distributions of returns for one-dimensional random walks, defined in discrete time and space, with jumps restricted to nearest neighbors and constant transition rates, are investigated. Explicit expressions are provided for certain parameter sets, and an integral representation is developed, with the asymptotic behavior rigorously characterized across different domain regions. It is shown that, in the symmetric case, the distribution of returns exhibits a universal asymptotic power-law decay with an exponent of -3/2, whereas deviations from symmetry introduce exponential corrections. These results are connected with nonadditive statistical mechanics by showing that a q-Gaussian fit would imply a nonadditively parameter q=7/3. The analysis provides an expression for the distribution of returns as a Stieltjes transform as well as a framework for understanding fat-tailed distributions of returns in discrete random walk models.