Efron's asymptotic monotonicity property in the Gaussian stable domain of attraction

Summary: This study examines linear fair risk sharing rules and conditional mean risk sharing rules for independent but heterogeneous losses in an insurance pool. It is found that insurance at pure premium can be obtained for an infinitely large pool, and the difference between actual contributions and pure premiums will ultimately follow a Normal distribution. Additionally, a linear fair risk sharing rule approximating the conditional mean risk sharing rule is identified, providing a practical simplification for use within large pools. Furthermore, the approximate number of participants needed to maintain the volatility of individual contributions within an acceptable range is determined based on the established asymptotic Normality.