## 4 Application  
### Identifying the Earthquake Interevent Time Correlations  
The detection of the correlations is investigated through time series analysis. Specifically, the _ACF_ and _PACF_ values, along with the statistics of Ljung-Box Q-test are sequentially applied for each \(\Delta T_{i}\) data sample (\(\Delta T_{1}\), \(\Delta T_{2}\),..., \(\Delta T_{n}\) for \(M_{w}\geq 4.1\), \(M_{w}\geq 4.2\),..., respectively) for a given sub-area until the temporal crossover between correlated and statistically independent events is determined. It is worth mentioning that the 95% confidence intervals of _ACF_ and _PACF_ are calculated separately for each data sample \(\Delta T_{i}\), which obtains a different length. Figures 3-5 illustrate the results of the correlation analysis for the \(\Delta T\) data set for the central Ionian Islands area (sub-area 2). The values of _ACF_ (Figure 3) exhibit strong positive autocorrelation until the magnitude threshold \(M_{thr}\geq 4.6\) (Figure 3a-f). From this \(M_{thr}\) onwards, the autocorrelation becomes weaker (Figure 3g_h) until \(M_{thr}\geq 4.9\) (Figure 3i) when no significant autocorrelations of any lag are recorded, and the relevant samples of \(\Delta T\) can be considered statistically uncorrelated. The values of _PACF_ (Figure 4) confirm this crossover between correlated and statistically independent events at the magnitude threshold of \(M_{thr}\geq 4.9\). \begin{table}
\begin{tabular}{c c c} \hline
**Distribution** & **Probability Density Function** & **Parameters** \\ \hline Gamma & \(f(x|\kappa,\theta)=\frac{1}{\rho\Gamma(x)}x^{x-1}\exp\left\{\frac{-x}{\theta}\right\}\) & \(\kappa\) (shape) \\  & \(\theta\) (scale) \\  & \(\mu\) (natural logarithm of mean value) \\  & \(\sigma\) (natural logarithm of standard deviation) \\  & \(\sigma\) (natural logarithm of standard deviation) \\  & \(\alpha\) (scale) \\  & \(b\) (shape) \\  & \(f(x|\mu)=\frac{1}{\mu}\exp\left\{-\frac{x}{\mu}\right\}\) & \(\mu\) (mean value) \\ \hline \end{tabular}
\end{table} 
Table 1: Definitions of the probability density functions (pdf) of the Gamma, Lognormal, Weibull and Exponential distributions applied in the statistically independent interevent times data, \(\Delta T\), of the 10 sub-areas of broader Aegean region. Figure 4: _PACF_ values (orange stems) and their 95% confidence intervals (blue lines) for central Ionian Islands (sub-area 2) for magnitude thresholds (\(M_{thr}\)) varying from 4.1 to 4.9 (subplots **a–i**). Figure 3: _ACF_ values (red stems) and their 95% confidence intervals (blue lines) for central Ionian Islands (sub-area 2) for magnitude thresholds (\(M_{thr}\)) varying from 4.1 to 4.9 (subplots **a–i**). For securing that the finding of the memory weakening as the magnitude threshold increases is not biased because of the progressive decrease in the data samples length, the computation of the number of lags exceeding the respective level of significance over the total number of lags for both the _ACF_ and _PACF_ values is implemented. The \(\Delta T\) data samples of this sub-region exhibit an increase in the ratio for the ACF values from 0.33 to 0.60 for the \(M_{thr}\geq 4.1\) and \(M_{thr}\geq 4.2\), respectively. From an \(M_{thr}\geq 4.3\) and above we observe a systematic decrease in this ratio for _ACF_ values. We obtained the same results for the ratio related to the respective values of _PACF_. Taking into account these results, it could be stated that the smaller length of the respective data samples as the \(M_{thr}\) values increase does not bias the results of the _ACF_ and _PACF_ calculations, implying weakening of the memory with respect to the increase in magnitude threshold. The results of the Ljung-Box Q-test (Figure 5) unveil the same temporal behavior of the \(\Delta T\) data samples.