Correlations between Rare Events for Gaussian Stochastic Processes with Long-Term Memory

Rare events refer to qualitatively unlikely events whose realization can nevertheless have important consequences. Typically, the prediction of the kinetics of these events relies on Arrhenius laws, with exponentially distributed waiting times, and no correlations between successive occurrences. However, this description breaks down in the presence of long-term memory, as has been observed in the contexts of geophysical time series or protein dynamics. So far, existing analytical approaches do not quantify the correlations between rare events due to long-term memory. Here, for non-Markovian Gaussian processes, we determine analytically the impact of long-term memory on the distribution of first and second passage times to a rarely reached threshold, using a perturbation approach. This distribution is nonexponential, thus going beyond the Arrhenius paradigm. We obtain an explicit expression for the covariance between the first and second passage times, and we predict how the mean time to the next extreme event depends on the previous passage time, illustrating the phenomenon of clustering of extreme events. These analytical results, validated through extensive stochastic simulations, shed lights on the strong correlation between successive occurrences of extreme events due to long-term memory.