## 4 Discussion: Application to the Current Probability in Italy  
An actual application is Italy. The historic Italian catalog CPT115 is probably the best one in the world; unfortunately, being the best does not ensure that it is good enough. It is very rich, but not complete, dating back to about 1600. Thus, it is quite obvious that we cannot use it at face value. However, since earthquakes are self-similar, we can try to prove that the worldwide survival probability general function can be applied to the Italian events. On the basis of DS6, I created \(10^{6}\) datasets with randomly picked magnitudes, according to a Gaussian distribution, based on the catalog magnitudes and relative standard deviation. The obvious assumption was that all (or most) of the M \(\geq\) 6 earthquakes are, in fact, contained in CPT15, but that the magnitudes are either over- or under-estimated. The randomized datasets contained between 55 and 92 events each. I discretized each logarithmic decade into 20 bins, instead of directly using the observed \(\Delta t\)s. Then, I computed the survival probability function for each dataset, stacked it, and finally averaged it. As the event number was not constant, the most significant portion of the averaged survival probability function was where events tended to concentrate, that is, in the \(\Delta t\) range between 0.1 and three. In Figure 8, a comparison between DS1, DS5, and DS6 is shown, where the solid line is DS5, the dotted line is DS1, and the dashed line is DS6. We can observe that the similarity between DS1 and DS6, in the range of interest, is quite remarkable, even though it is also acceptable with respect to DS5. This means that, if we have a poor dataset, we can substitute it with a richer one with the same magnitude completeness, as long as we know precisely the timing of the last two events and have a realistic estimate of the event rate. Evaluation of hazard is tricky from a computational viewpoint; that is, the application of what is shown in Figure 7 requires some further steps. First of all, we need to oversample the survival probability function which we will use (i.e., we need to discretize each logarithmic decade into at least 200 bins), and then we need to apply a smoothing technique, as the function would otherwise assume a step-like shape. I personally found that a Gaussian filter in the time domain with \(\sigma=0.2\) had an optimal response, since it ensures monotonic behavior (except for the very few initial points) and minimal differences with respect to the original function. However, I applied this filter not to the time domain function itself, but to its equivalent in the time logarithmic domain. As, in the very few initial points, it yielded a strong down-shoot followed by a very slight over-shoot, which brings the probability to over one; I included an initial logarithmic decade and padded it with ones. Finally,  
Figure 7: Method of computation of the 10% probability time interval at the elapsed time \(t^{*}\), given a specific \(\Delta t\) between the last and the second-to-last event. The solid line refers to the empirical function, the dashed line refers to the Poissonian one, and the dotted line refers to the new, rescaled empirical one. a specific \(t^{*}\) generally requires interpolation between two samples, as it will follow the 90% threshold. I applied a simple linear interpolation technique, but more sophisticated approaches will yield a more refined result. The final step is to select the optimal bandwidth over which to compute the survival probability function. As my main reference dataset was DS1 (which has nearly 7000 events), I chose a band including the nearest 250 data points on both sides (500 total) of the specific \(\Delta t\) value between the last and second-to-last events. In the case that this \(\Delta t\) is either very small or very large and I do not have 250 events on both sides, I used the smallest or largest 500 \(\Delta t\), respectively.