{"ID":2874376,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.05497","arxiv_id":"2509.05497","title":"Broadly discrete stable distributions","abstract":"Stable distributions are of fundamental importance in probability theory, yet their absolute continuity makes them unsuitable for modeling count data. A discrete analog of strict stability has been previously proposed by replacing scaling with binomial thinning, but it only holds for a subset of the tail index parameters. Here, we generalize the discrete stable class to the full range of tail indices and show that it is equivalent to the mixed Poisson-stable family. This broadly discrete stable family is discretely infinitely divisible, with a compound Poisson representation involving a novel generalization of the Sibuya distribution. Under additional parameter constraints, they are also discretely self-decomposable and unimodal. The discrete stable distributions provide a new frontier in probabilistic modeling of both light and heavy tailed count data.","short_abstract":"Stable distributions are of fundamental importance in probability theory, yet their absolute continuity makes them unsuitable for modeling count data. A discrete analog of strict stability has been previously proposed by replacing scaling with binomial thinning, but it only holds for a subset of the tail index paramete...","url_abs":"https://arxiv.org/abs/2509.05497","url_pdf":"https://arxiv.org/pdf/2509.05497v1","authors":"[\"F. William Townes\"]","published":"2025-09-05T21:09:46Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\"]","methods":"[]","has_code":false}