On a Tail Bound for Root-Finding in Randomly Growing Trees

2019/05/18 by Sam Justice, Justice, Sam, Nariankadu D. Shyamalkumar +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1905.07652

openalex publication_date 2019/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We re-examine a lower-tail upper bound for the random variable X=∏_i=1^∞min\∑_k=1ⁱE_k,1\, where E₁,E₂,…\stackreliid\simExp(1). This bound has found use in root-finding and seed-finding algorithms for randomly growing trees, and was initially proved as a lemma in the context of the uniform attachment tree model. We first show that X has a useful representation as a compound product of uniform random variables that allows us to determine its moments and refine the existing nonasymptotic bound. Next we demonstrate that the lower-tail probability for X can equivalently be written as a probability involving two independent Poisson random variables, an equivalence that yields a novel general result regarding indpendent Poissons and that also enables us to obtain tight asymptotic bounds on the tail probability of interest.

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