#### 3.1. Exploratory Data Analysis

Our time series does not show significant inhomogeneities, according to the Buishand range test [

40], which detects break years in annual data (

Figure 2a).

The frequency distribution approximates Gaussian-like distributional features (

Figure 2b), while the attractor in the time-state domain shows a ramification towards the diagonal arrow (

Figure 2c), which indicates a possible predictable pattern (whether reasonably accurate forecasts can be made).

The Hurst exponent estimated by the R/S method [

41,

42] is equal to 0.600, which is near the threshold of 0.65 used by Quian and Rasheed [

43] to identify a series that can be predicted accurately. However, the Hurst exponent >0.50 indicates that some dependence structure exists, which advocates the foreseeability of the original series.

The model returns a periodical pattern in the extreme rainfall evolution with important cycles equal to 54, which imposes a training period of at least 109 years (as from the software Smooth Forecast). A range of four decades is the maximum allowed for an interdecadal perspective, leaving 111 years available for training (from 1866 to 1976).

#### 3.2. Validation Test

For the validation period between 1977 and 2016 (

Figure 3a), the simulation results are quite promising, judging by the overall closeness of the red prediction curve to the blue curve of observed extreme rainfall evolution.

For the validation period, the metric statistics R, RMSE, MAPE, and MASE, which are equal to 0.40, 25 mm, 26.5%, and 0.692, respectively, indicate a satisfactory performance, and imply that the forecast model is superior to a random walk. Model residuals have a quasi-normal distribution (

Figure 3b), and the Q-Q (quantile-quantile) plot shows that the theoretical and sample quantiles are quite similar (

Figure 3c). Owing to only a few data points, the distribution is slightly right-skewed, rising more slowly at the beginning and then rising at a faster rate for higher values.

Overall, the model provides realistic values on average, but peaks are often not captured. The results thus indicate that the exponential model for rainfall simulation can be employed to reasonably characterize and predict dominant patterns of variability at interannual to interdecadal time scales, while they may not be unambiguously disclosed in short-term (one year ahead or so) forecasting.

#### 3.3. Simulation Experiment

The ability of the model to extrapolate results is dependent on the stochastic and deterministic behaviours of time-dependent terms. In this specific case, the system appears to evolve as influenced by natural variations, which are also weakly predictable (after Miglietta et al. [

44]).

When examining the projection of daily maximum rainfall over the four future decades (2017–2066,

Figure 4), the values of the ensemble (black curve) are observed to roughly lie around or above the long-term median (bold grey lines). The plumes band (colour curves) gives an approximate idea of the uncertainty associated with the forecasts. A weak increasing trend is shown (

Figure 5), by almost the same pace as it was in the previous three decades.

The extreme value of about 130 mm, corresponding to a 100-year return period (red line), is overstepped one time within the forecasted period. Also, the interdecadal variability is pronounced, with the annual single values often approaching the 10-year return period of 93 mm (orange line), as estimated by GEV (generalized extreme value) distribution (

Figure 6a).

The observable trend into the projection is crossed by cyclical patterns that are still present as in the past, with a moderate magnitude around the 2030s and 2050s decades.

Figure 4 also displays the oscillation of the mean of the plumes forecast (with peaks after 2050), which encloses the CMIP5 (Coupled Model Intercomparison Project Phase 5) smoothed mean (blue dots). Thus, the latter shows a similarity in the cyclical pattern with our projections. However, the general trend of the CMIP5 output is approaching 0.04 mm yr

^{−1}, which is about four times higher than that of the exponential smoothing forecast (

Figure 5).

This appears compatible with an intensification of extreme precipitation events during the 21st century, which are attributed to the increasing atmospheric moisture content in a warming climate (according to the Clausius-Clapeyron equation) and projected by GCMs modulated by a range of factors such as temperature lapse rate, vertical wind velocities, and temperature anomalies when extreme events occur [

45,

46]. These smoothed oscillations may be induced by atmospheric and ocean forcing, such as the Pacific Decadal Oscillation (PDO) and the Atlantic Multidecadal Oscillation (AMO) that act on the hydroclimatic system in multiple ways. Indeed, the studied time series appears to be just cross-correlated with teleconnection indices such as the PDO and AMO (

Figure 6b). This cross-correlation supports the predictability of our model, implying that the statistical model would reproduce a coupled oscillation between extreme rainfalls and AMO–PDO indices. Our analysis (

Figure 7) does not support a relation between our output (extreme rainfall at Naples, mostly occurring in autumn) and dynamics of the North Atlantic Oscillation (NAO), whose major influence on Italian precipitation (mostly winter precipitation) is documented [

47]. Daily maximum rainfall in Naples is only weakly correlated with NAO (

Figure 7a), while the combined AMO/PDO index shows a comparable course (

Figure 7b). In particular, higher (lower) rainfall maxima tends to coincide with positive (negative) phases of the combined index. Knudsen et al. [

48] conjectured that a quasi-persistent ~55-to-70-year AMO, linked to internal ocean–atmosphere variability, existed during large parts of the Holocene, thus suggesting that the coupling of AMO and regional climate conditions was modulated by orbitally-induced shifts in large-scale ocean–atmosphere circulation.

These results suggest an astronomic origin of the ~50-to-70-year variability found in several climatic records [

49]. Moreover, the PDO can modulate the interannual relationship between El Niño Southern Oscillation and the global climate, with large dry–wet variations over northern Europe and the Mediterranean [

50]. The PDO influence can be transmitted to the Atlantic European region indirectly via the NAO by the mechanism involving a shift of the North Atlantic storm track [

51]. In this way, cyclone activity tends to enhance during the positive AMO phases and the negative PDO phases, when the zonal flow is further south.