Tsallis statistics and gradually truncated Lévy flight—distribution of an economical index

Tsallis postulated a generalized form for entropy and give rise to a new statistics now known as Tsallis statistics. In the present work, we compare the Tsallis statistics with the gradually truncated Lévy flight, and discuss the distribution of an economical index—the Standard and Poorʹs 500—using the values of standard deviation as calculated by our model. We find that both statistics give almost the same distribution. Thus we feel that gradual truncation of Lévy distribution, after certain critical step size for describing complex systems, is a requirement of generalized thermodynamics or similar. The gradually truncated Lévy flight is based on physical considerations and bring a better physical picture of the dynamics of the whole system. Tsallis statistics gives a theoretical support. Both statistics together can be utilized for the development of a more exact portfolio theory or to understand better the complexities in human and financial behaviors. A comparison of both statistics is made.