## 5 Conclusions  
The detection of the threshold magnitude, \(M_{thr}\), above which the earthquake interevent times, \(\Delta T\), might be considered statistically independent, is investigated through the _ACF_ and _PACF_ values along with the application of the Ljung-Box Q-test in 10 distinctive sub-areas of the Greek territory. The analysis revealed that in all cases the results of the Ljung-Box Q-test adequately agree with the results derived by _ACF_ and _PACF_. Detrended Fluctuations Analysis results further confirm the weakening of the memory as the \(M_{thr}\) of earthquake interevent time data increases. The obtained temporal crossover of the \(\Delta T\) data ranges from \(M_{thr}\geq 4.7\) up to \(M_{thr}\geq 5.1\) in the 10 sub-areas. These findings are statistically very important because one can determine above which magnitude threshold the property of mutual independence is fulfilled. This independency property concerns either the events occurring in two different time intervals or only the occurrence time of the last event but not the full seismicity history. The proposed workflow for the investigation of the crossover behavior of \(\Delta T\) data is easy to apply and capable of providing reliable results aimed at fixing the methodological issue of statistical independence. In other words, it could be considered as a routine initial step before a certain stochastic model application in earthquake interevent time data. Once the magnitude thresholds above which the \(\Delta T\) data samples are statistically independent are identified, the Gamma, Weibull, Lognormal and Exponential distributions are applied and compared in the respective data samples in order to model their temporal behavior. The comparison, which was implemented via the combination of the A-D goodness of fit test and the values of AIC and BIC criteria, revealed that the Gamma distribution exhibits the best performance. This latter result agrees with the analysis performed by [PERSON] [7,8], who claimed that the earthquake interevent times follow a universal scaling law which could be represented by the Gamma distribution. This fact implies that the \(\Delta T\) can be described better with a general renewal process rather than the Markovian Poisson process. The values of the shape parameter, \(k\), of Gamma distribution, which characterizes the seismic process, in all cases are estimated as \(k<1\) implying a clustering behavior of seismicity. In such cases, the earthquake occurrence probability soon after a strong earthquake is high, and most events occur at times less than the mean interevent time. It is notable that sub-areas with low seismic activity are associated with larger values of \(k\). In more detail, the largest estimated value of \(k\) is equal to 0.54 in the southwestern Aegean Sea and Peloponnese (sub-area 6), an area with considerably lower seismicity than the Corinth Gulf (sub-area 7) and North Aegean Sea (sub-area 9), in which the estimated \(k\) values are equal to 0.27 and 0.25, respectively. The smallest value of \(k\) (\(k=0.22\)) is estimated in the central Ionian Islands (sub-area 2), which exhibits the highest moment rate in the Mediterranean region. This study's results provide information on the temporal behavior of the \(\Delta T\) of moderate to large earthquakes over a sufficiently long period (1975-2021). These findings might be potentially used for both additional statistical applications (e.g., Stress Release applications), in which the need for documenting the independency magnitude cut-off is necessary, and as initial earthquake occurrence forecast models in intermediate time scales, which can be considered as inputs for improving regional seismic hazard assessment investigations (e.g., [46]). Conceptualization, C.K., G.T., E.P. and V.K. ; data curation, C.K., E.P. and V.K. ; formal analysis, C.K., G.T., E.P. and V.K. ; investigation, C.K. ; methodology, C.K. and G.T. ; Validation, C.K. ; writing--original draft, C.K. ; writing--review and editing, C.K., G.T., E.P. and V.K.