math.PR — Probability

Classical infinite divisibility, self-decomposability and bell-shape of free stable laws

Contributed by Min WANG

We determine the classical infinite divisibility, self-decomposability and extended Thorin regions of the strictly free stable family. Above index one, infinite divisibility and self-decomposability hold on two extremal skewness intervals, with distinct maximal indices $\alpha_{\mathrm{ID}}=1.5240739610\ldots$ and $\alpha_{\mathrm{SD}}=1.4283571142\ldots$. Below one, every law is self-decomposable, whereas the Thorin region is a central interval; the optimal uniform bound is $4/5$. We characterize the boundaries by global contact conditions and derive their different asymptotic scales at index one. A common phase representation also gives an exact bell-shape test, strictly bell-shaped examples outside the Thorin class, and an unbounded number of skewness components near one. The exceptional free one-stable family admits an explicit integer criterion. The proofs combine analytic estimates with finite interval certificates. The bell-shape results do not constitute a full parameter classification.