Boundedness of discounted branching random walks via generic chaining

Consider a discrete-time supercritical discounted branching random walk, in which increments at depth $k$ are independent and identically distributed with the same law as $m^{-kH}Y$, where $Y$ has a fixed law, $H>0$, and $m>1$ is the expected number of offspring at depth one. We provide a clean characterization of the boundedness of the discounted branching random walk: under mild conditions on the offspring distribution, the process is almost surely bounded if and only if $\mathbb{E}[|Y|^{1/H}]<\infty$. This extends results of Athreya (1985) and A\"id\'ekon--Hu--Shi (2024), and provides a partial answer to Open Problem 31 of Aldous--Bandyopadhyay (2005).