Crossing intervals of non-Markovian Gaussian processes

We review the properties of time intervals between the crossings at a level $M$ of a smooth stationary Gaussian temporal signal. The distribution of these intervals and the persistence are derived within the independent interval approximation (IIA). These results grant access to the distribution of extrema of a general Gaussian process. Exact results are obtained for the persistence exponents and the crossing interval distributions, in the limit of large $\mid M\mid$. In addition, the small-time behavior of the interval distributions and the persistence is calculated analytically, for any $M$. The IIA is found to reproduce most of these exact results, and its accuracy is also illustrated by extensive numerical simulations applied to non-Markovian Gaussian processes appearing in various physical contexts.

Figure 1

(Color online) ${\theta }_{-}\left(M\right)$ (in unit of ${\tau }_0^{-1}$), for the non-Markovian process associated with the correlator $f_2\left(t\right)$ of Eq. (88). Symbols are the results of numerical simulations [see text and Eq. (104)], while the full line is the result of the IIA approximation, Eq. (97). Finally, the dashed line corresponds to the exact asymptotic result, ${\theta }_{-}\left(M\right)\sim {\tau }_{-}^{-1}$. Some values of ${\theta }_{-}\left(M\right)$ are also reported in Table .

Figure 2

(Color online) Distribution of $\epsilon =\pm$ intervals, $P_{\pm }\left(t\right)$, for $M=3$, as a function of $t∕{\tau }_{\pm }$ (full lines for $M=3$, ${\tau }_{+}\approx 0.76350a_2^{-1∕2}$, and ${\tau }_{-}\approx 564.83a_2^{-1∕2}$), for the process associated with the correlator $f_2\left(t\right)$ of Eq. (88). The distribution of − intervals has converged to the Poissonian form of Eq. (119) (dashed line). Note that the linear regime of $P_{-}\left(t\right)$ predicted in Eqs. (163, 168), for $t⪡{\tau }_{+}⪡{\tau }_{-}$, cannot be seen at this scale. The distribution of + intervals is well described by the Wigner distribution of Eq. (122) (dashed line), but ultimately decays exponentially for $t⪢a_2^{-1∕2}$ (inset: dotted line of slope ${\tau }_{+}{\theta }_{+}\approx 2.35$).

Figure 3

(Color online) ${\theta }_{+}\left(M\right)$ (in unit of ${\tau }_0^{-1}$) for the non-Markovian process associated with the correlator $f_2\left(t\right)$ of Eq. (88). Symbols correspond to results of numerical simulations. The upper and lower dashed lines are the exact bounds of Eq. (150) (the upper bound being exact up to order $M$, for small $M$). The ratio of these two bounds goes to 1 when $M\to +\infty$. The full line is a quadratic fit for $M>3∕2$, to the functional form ${\theta }_{+}\left(M\right)=a_0+a_{1}M+[M^2∕4\widehat{f}\left(0\right)]$, with $a_0\approx 2.0$ and $a_1\approx 0.66$. Finally, the middle dashed line is the IIA result [along with its asymptotic slope given by Eq. (136); dotted line], which underestimates the quadratic growth of ${\theta }_{+}\left(M\right)$. Some values of ${\theta }_{+}\left(M\right)$ are also reported in Table .

Figure 4

(Color online) Distribution of $\epsilon =\pm$ intervals, $P_{\pm }\left(t\right)$, for $M=1∕2$, as a function of $t∕{\tau }_{\pm }$ (full lines), for the process associated with the correlator $f_2\left(t\right)$ of Eq. (88) [$P_{-}\left(t\right)$ is the most peaked distribution]. The straight dotted lines have the predicted slopes at $t=0$, given by Eqs. (163, 165, 168). The dashed lines are the distributions obtained from the IIA, after taking the inverse Laplace transform of Eqs. (76, 77). The inset shows the same data on a semi logarithmic scale, illustrating the good accuracy of the IIA in predicting the persistence exponents ${\theta }_{\pm }\left(M\right)$ and their associated residue $R_{\pm }$.