First-Passage Time Properties of Correlated Time Series with Scale-Invariant Behavior and with Crossovers in the Scaling

Abstract

The observable outputs of a great variety of complex dynamical systems form long-range correlated time series with scale invariance behavior. Important properties of such time series are related to the statistical behavior of the first-passage time (FPT), i.e., the time required for an output variable that defines the time series to return to a certain value. Experimental findings in complex systems have attributed the properties of the FPT probability distribution and the FPT mean value to the specifics of the particular system. However, in a previous work we showed (Carretero-Campos, Phys Rev E 85:011139, 2012) that correlations are a unifying factor behind the variety of findings for FPT, and that diverse systems characterized by the same degree of correlations in the output time series exhibit similar FPT properties. Here, we extend our analysis and study the FPT properties of long-range correlated time series with crossovers in the scaling, similar to those observed in many experimental systems. To do so, first we introduce an algorithm able to generate artificial time series of this kind, and study numerically the statistical properties of FPT for these time series. Then, we compare our results to those found in the output time series of real systems and we demonstrate that, independently of the specifics of the system, correlations are the unifying factor underlying key FPT properties of systems with output time series exhibiting crossovers in the scaling.

Appendix

For brevity, we only include here the derivation of the behavior of $\langle \ell\rangle$ as a function of t _c for the case α _s, α _l > 1, shown graphically in Fig. 4b. For such case, the functional forms corresponding to both exponents are power-laws, and the transition between them occurs at $g(t_{\mathrm{c}}) \sim t_{\mathrm{c}}$. Thus, p(ℓ) is of the form:

$$\displaystyle{ p(\ell) = k\left \{\begin{array}{ccc} \ell^{-(3-\alpha _{\mathrm{s}})} & \mathrm{if}& \ell \leq t_{\mathrm{c}} \\ \frac{t_{\mathrm{c}}^{3-\alpha _{\mathrm{l}}}} {t_{\mathrm{c}}^{3-\alpha _{\mathrm{s}}}} \ell^{-(3-\alpha _{\mathrm{l}})} & \mathrm{if}&t_{\mathrm{ c}} <\ell<N \end{array} \right. }$$

(7)

where k is a normalization constant that can be obtained from $\int _{1}^{N}p(\ell)\,d\ell = 1$, and the factor $t_{\mathrm{c}}^{3-\alpha _{\mathrm{l}}}/t_{\mathrm{ c}}^{3-\alpha _{\mathrm{s}}}$ ensures continuity at t _c. The mean FPT value is then

$$\displaystyle{ \langle \ell\rangle =\int _{ 1}^{N}\ell\,p(\ell)\,d\ell = \frac{\frac{t_{\mathrm{c}}^{\alpha _{\mathrm{s}}-1}-1} {1-\alpha _{\mathrm{s}}} \ + \frac{N^{\alpha _{\mathrm{l}}-1}t_{\mathrm{ c}}^{3-\alpha _{\mathrm{l}}}-t_{\mathrm{ c}}^{2}} {t_{\mathrm{c}}^{3-\alpha _{\mathrm{s}}}(1-\alpha _{\mathrm{l}})} } {\frac{t_{\mathrm{c}}^{\alpha _{\mathrm{s}}-2}-1} {2-\alpha _{\mathrm{s}}} + \frac{N^{\alpha _{\mathrm{l}}-2}t_{\mathrm{c}}^{3-\alpha _{\mathrm{l}}}-t_{\mathrm{c}}} {t_{\mathrm{c}}^{3-\alpha _{\mathrm{s}}}(2-\alpha _{\mathrm{l}})} } }$$

(8)

This expression is complicated, but noting that 1 < α _s, α _l < 2, we can consider the limit of large time series length N by keeping only the largest powers of N in the numerator and denominator of (8). Similarly, and as we are interested in the behavior of $\langle \ell\rangle$ as t _c increases, we can keep only the highest powers of t _c in the numerator and denominator of the result. Altogether, we obtain:

$$\displaystyle{ \langle \ell\rangle \simeq \frac{2 -\alpha _{\mathrm{s}}} {\alpha _{\mathrm{l}} - 1}N^{\alpha _{\mathrm{l}}-1}t_{\mathrm{c}}^{\alpha _{\mathrm{s}}-\alpha _{\mathrm{l}}} }$$

(9)

This equation is in perfect agreement with the numerical results shown in Fig. 4b: for increasing t _c values, $\langle \ell\rangle$ changes between its two extreme values as a power-law of t _c with exponent α _s −α _l (dotted lines in Fig. 4b). Also, Eq. (9) shows the dependence of $\langle \ell\rangle$ on the time series length N in the limit of large N for fixed t _c. In this case, and as t _c is finite, the function p(ℓ) (7) is governed in the range $(t_{\mathrm{c}},\infty )$ by the scaling exponent α _l, and this range controls the mean value ($\langle \ell\rangle \sim N^{\alpha _{\mathrm{l}}-1}$) exactly in the same way as in the case of time series with single scaling exponents with 1 < α < 2 [see Eq. (3)].