Threshold dynamics in age-structured distributions with expanding support: A unified mathematical framework

Threshold ages — the ages at which mortality reductions shift from compressing to expanding lifespan variation — are fundamental features of age-at-death distributions. Classical continuous formulations typically derive threshold conditions on the half-line [0,∞); empirical life tables and computational implementations, however, operate on a finite, maximum observed age ω that can expand over time. We develop a unified framework for threshold dynamics under expanding support. First, we show that κ-homogeneous spread measures admit a scale–shape decomposition, separating the mechanical effect of domain size from distributional shape. Second, differentiating this decomposition yields a sensitivity decomposition into a shape channel and a domain-scaling channel induced by changes in ω. The latter shifts thresholds toward younger ages for all κ>0. Third, we prove existence, uniqueness, and a general threshold-shift formula, expressed through the normalized threshold y∗=x∗/ω. Applications to variance, the Gini coefficient, entropy, and higher moments, together with simulations and Human Mortality Database data, confirm the theoretical predictions. In protected, predominantly senescent populations, normalized thresholds cluster in narrow ranges, but historical analysis shows that y∗ can evolve substantially during demographic transitions (e.g., for Swedish females, the variance threshold shifts from yV∗≈0.45 in 1751 to yV∗≈0.75 in 2020).