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The normal distribution is used as a unified probability distribution, however, our researcher found that it is not good agreed with the real-life dynamical systems data. We collected and analyzed representative naturally occurring data series (e.g., the earth environment, sunspots, brain waves, electrocardiograms, some cases are classic chaos systems and social activities). It is found that the probability density functions (PDFs) of first or higher order differences for these datasets are consistently fat-tailed bell-shaped curves, and their associated cumulative distribution functions (CDFs) are consistently S-shaped when compared to the near-straight line of the normal distribution CDF. It is proved that this profile is not because of numerical or measure error, and the t-distribution is a good approximation. This kind of PDF/CDF is a universal phenomenon for independent time and space series data, which will make researchers to reconsider some hypotheses about stochastic dynamical models such as Wiener process, and therefore merits investigation.
Electric signals have been recently recorded at the Earths surface with amplitudes appreciably larger than those hitherto reported. Their entropy in natural time is smaller than that, $S_u$, of a ``uniform distribution. The same holds for their entro
We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of th
In high frequency financial data not only returns but also waiting times between trades are random variables. In this work, we analyze the spectra of the waiting-time processes for tick-by-tick trades. The numerical problem, strictly related with the
When both the difference between two quantities and their individual values can be measured or computational predicted, multiple quantities can be determined from the measurements or predictions of select individual quantities and select pairwise dif
In this paper we present the concept of description of random processes in complex systems with the discrete time. It involves the description of kinetics of discrete processes by means of the chain of finite-difference non-Markov equations for time