2018/04/20 by Marcus Michelen, Julian Sahasrabudhe, Michelen, Marcus +1
Mathematics · #26C05 #26C10 #60F05 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1804.07696
openalex publication_date 2018/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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