Extreme events, which are usually characterized by generalized extreme value models, can exhibit long-term memory, whose impact needs to be quantified. It was known that extreme recurrence intervals can better characterize the significant influence of long-term memory than using the GEV model. Our statistical analyses based on time series datasets following the Lévy stable distribution confirm that the stretched exponential distribution can describe a wide spectrum of memory behavior transition from exponentially distributed intervals to power-law distributed ones, extending the previous (...) evaluation of the stretched exponential function using Gaussian/exponential distributed random data. Further deviation and discussion of a historical paradox are also provided, based on the theoretical analysis of the Bayesian law and the stretched exponential distribution. (shrink)