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

05/18/2019

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by   Sam Justice, et al.

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We re-examine a lower-tail upper bound for the random variable X=∏_i=1^∞{∑_k=1^iE_k,1}, where E_1,E_2,...iid∼Exp(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.