{"paper":{"title":"Central limit theorems from the roots of probability generating functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.PR","authors_text":"Julian Sahasrabudhe, Marcus Michelen","submitted_at":"2018-04-20T15:56:00Z","abstract_excerpt":"For each $n$, let $X_n \\in \\{0,\\ldots,n\\}$ be a random variable with mean $\\mu_n$, standard deviation $\\sigma_n$, and let \\[ P_n(z) = \\sum_{k=0}^n \\mathbb{P}( X_n = k) z^k ,\\] be its probability generating function. We show that if none of the complex zeros of the polynomials $\\{ P_n(z)\\}$ are contained in a neighbourhood of $1 \\in \\mathbb{C}$ and $\\sigma_n > n^{\\varepsilon}$ for some $\\varepsilon >0$, then $ X_n^* =(X_n - \\mu_n)\\sigma^{-1}_n$ tends to a normal random variable $Z \\sim \\mathcal{N}(0,1)$ in distribution as $n \\rightarrow \\infty$. Moreover, we show this result is sharp in the sen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.07696","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}