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Scaling and memory in seismological phenomena

Abe Sumiyoshi, Suzuki Norikazu

Acta Geophysica

2023

Vol. 71, no. 5

2081--2087

The concept of memory is of central importance for characterizing complex systems and phenomena. Presence of long-term memories indicates how their dynamics can be less sensitive to initial conditions compared to the chaotic cases. On the other hand, it is empirically known that the Feller–Pareto distribution, which decays as the power law i.e. the scale-invariant nature, frequently appears as a statistical law generated by the dynamics of complex systems. However, it is generally not a simple task to determine if a system obeying such a power law possesses a high degree of complexity with a long-term memory. Here, a new method is proposed for characterization of memory. In particular, a scaling relation to be satisfied by any memoryless dynamics generating the Feller–Pareto power-law distribution is presented. Then, the method is applied to the real data of energies released by a series of earthquakes and acceleration of ground motion due to a strong earthquake. It is shown in this way that the sequence of the released energy in seismicity is memoryless in the event time, whereas that of acceleration is memoryful in the sampling time.

Limits of truncation experiments

Marohn F.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

71--88

Given n i.i.d. copies X1,…., Xn of a random variable X with distribution Pѵ, ѵ θ∈Θ⊂Rk, we are only interested in those observations that fall into some set D = D(n) ⊂ Rd having but a small probability of occurrence. The truncation set D is assumed to be known and non-random. Denoting the distribution of the truncated random variable X1D(X) by Pnѵ we consider the triangular array of experiments (Rd, ßd, (Pnѵθ ), n∈N, and investigate the asymptotic behavior of the n-fold products ((Rd)n, (ßd)n, (Pnѵ)nѵ∈Θ ). Under a suitable density expansion,Gaussian shifts as well as Poisson experiments occur in the limit, where the latter case typically occurs when the number of expected observations falling in D is bounded. Finally, we investigate invariance properties of the occurring Poisson limits.