Understanding the nature of the long-range memory phenomenon in socioeconomic systems

In the face of the upcoming 30th anniversary of econophysics, we review our contributions and other related works on the modeling of the long-range memory phenomenon in physical, economic, and other social complex systems. Our group has shown that the long-range memory phenomenon can be reproduced using various Markov processes, such as point processes, stochastic differential equations and agent-based models. Reproduced well enough to match other statistical properties of the financial markets, such as return and trading activity distributions and first-passage time distributions. Research has lead us to question whether the observed long-range memory is a result of actual long-range memory process or just a consequence of non-linearity of Markov processes. As our most recent result we discuss the long-range memory of the order flow data in the financial markets and other social systems from the perspective of the fractional L\`{e}vy stable motion. We test widely used long-range memory estimators on discrete fractional L\`{e}vy stable motion represented by the ARFIMA sample series. Our newly obtained results seem indicate that new estimators of self-similarity and long-range memory for analyzing systems with non-Gaussian distributions have to be developed.