Statistical Characteristics, Occurrence Probabilities, and Extreme-event Recurrence Analysis of Solar Proton Events during 1976–2026

Using long-term observations of >10 MeV solar proton events (SPEs) during 1976–2026, we investigate their temporal evolution, occurrence probability distributions, and extreme-event recurrence characteristics. The results show that SPE occurrence is strongly correlated with the solar activity cycle, with most events concentrated during the ascending and maximum phases, while Solar Cycle 23 exhibits the highest activity level. The integral occurrence probabilities are well described by power-law functions with exponential cutoffs, whereas the differential distributions approximately follow exponential decay relations, indicating that SPEs exhibit both scale-free behavior and high-energy suppression. The occurrence probabilities of different SPE classes (S1–S4) show asymmetric temporal distributions over the solar cycle, with probability peaks near solar maximum. Extreme value theory (EVT), including the block maxima (BM) and peaks over threshold (POT) approaches, is further applied to estimate the return levels of extreme events. The results suggest that centennial-scale extreme SPEs may reach peak proton fluxes on the order of 10 5 pfu, while millennial-scale events could exceed several 10 5 pfu. Among the investigated methods, the BM annual approach shows the best agreement with the long-term statistical fitting results, whereas model uncertainties increase significantly with recurrence time. These findings are important for understanding the statistical behavior of solar energetic particles, assessing extreme space-weather risks, and modeling radiation environments for future deep-space exploration missions.