The gradually truncated Lévy flight: stochastic process for complex systems

Power-law distributions, i.e. Lévy flights have been observed in various economical, biological, and physical systems in high-frequency regime. These distributions can be successfully explained via gradually truncated Lévy flight (GTLF). In general, these systems converge to a Gaussian distribution in the low-frequency regime. In the present work, we develop a model for the physical basis for the cut-off length in GTLF and its variation with respect to the time interval between successive observations. We observe that GTLF automatically approach a Gaussian distribution in the low-frequency regime. We applied the present method to analyze time series in some physical and financial systems. The agreement between the experimental results and theoretical curves is excellent. The present method can be applied to analyze time series in a variety of fields, which in turn provide a basis for the development of further microscopic models for the system.