Descending from infinity: convergence of tailed distributions.

We investigate the relaxation of long-tailed distributions under stochastic dynamics that do not support such tails. Linear relaxation is found to be a borderline case in which long tails are exponentially suppressed in time but not eliminated. Relaxation stronger than linear suppresses long tails immediately, but may lead to strong transient peaks in the probability distribution. We also find that a δ-function initial distribution under stronger than linear decay displays not one but two different regimes of diffusive spreading.