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We re-examine a lower-tail upper bound for the random variable$$X=\prod_{i=1}^{\infty}\min\left\{\sum_{k=1}^iE_k,1\right\},$$ where$E_1,E_2,\ldots\stackrel{iid}\sim\text{Exp}(1)$. This bound has found use inroot-finding and seed-finding algorithms for randomly growing trees, and wasinitially proved as a lemma in the context of the uniform attachment treemodel. We first show that $X$ has a useful representation as a compound productof uniform random variables that allows us to determine its moments and refinethe existing nonasymptotic bound. Next we demonstrate that the lower-tailprobability for $X$ can equivalently be written as a probability involving twoindependent Poisson random variables, an equivalence that yields a novelgeneral result regarding indpendent Poissons and that also enables us to obtaintight asymptotic bounds on the tail probability of interest.

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