2018/04/20 by Marcus Michelen, Julian Sahasrabudhe, Michelen, Marcus +1 · 1 citation
Mathematics · #26C05 #26C10 #60F05 #Classical Analysis and ODEs (math.CA) #Combinatorics #Combinatorics (math.CO) #Conjecture #Distribution (mathematics) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Meromorphic and Entire Functions #Physics #Probability (math.PR) #Quantum mechanics #Random variable #Sigma #Statistics #math.CA #math.CO #math.PR #msc:26C05 #msc:26C10 #msc:60F05
paper · pdf · doi:10.48550/arxiv.1804.07696
published in arXiv (Cornell University) (Cornell University) · Some more history added to the introduction
openalex publication_date 2018/04/20 · arxiv created 2018/06/12 · arxiv updated 2018/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For each n, let Xn \∈ 0,\…,n be a random variable with mean\n\μn, standard deviation \σn, and let \Pn(z) =
sumk=0n\n
mathbbP( Xn = k) zk , be its probability generating function. We show\nthat if none of the complex zeros of the polynomials Pn(z) are\ncontained in a neighbourhood of 1 \∈ \ℂ and \σn >\nn\ε for some \ε >0, then Xn^* =(Xn -\n\μn)\σ-1n tends to a normal random variable Z \∼\n\N(0,1) in distribution as n \→ \∞. Moreover, we show\nthis result is sharp in the sense that there exist sequences of random\nvariables Xn with \σn > C\log n for which Pn(z) has no roots\nnear 1 and Xn^* is not asymptotically normal. These results disprove a\nconjecture of Pemantle and improve upon various results in the literature. We\ngo on to prove several other results connecting the location of the zeros of\nPn(z) and the distribution of the random variables Xn.\n