On a tail bound for analyzing random trees

Sam Justice and N.D. Shyamalkumar

Statistics & Probability Letters, 2020, vol. 162, issue C

Abstract: We re-examine a lower-tail upper bound for the random variable X=∏i=1∞min∑k=1iEk,1,where E1,E2,…∼iidExp(1). This bound has found use in root-finding and seed-finding algorithms for randomly growing trees, and was initially established 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 independent Poissons and that also enables us to obtain tight asymptotic bounds on the tail probability of interest.

Keywords: Tail bound; Randomly growing tree; Poisson process; Independent Poissons (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1016/j.spl.2020.108746

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