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On a tail bound for analyzing random trees

Author

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  • Justice, Sam
  • Shyamalkumar, N.D.

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.

Suggested Citation

  • Justice, Sam & Shyamalkumar, N.D., 2020. "On a tail bound for analyzing random trees," Statistics & Probability Letters, Elsevier, vol. 162(C).
    • Handle: RePEc:eee:stapro:v:162:y:2020:i:c:s0167715220300493
    DOI: 10.1016/j.spl.2020.108746
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