Maximums on trees

We study the minimal/endogenous solution RR to the maximum recursion on weighted branching trees given by R=D(⋁i=1NCiRi)∨Q, where (Q,N,C1,C2,…)(Q,N,C1,C2,…) is a random vector with N∈N∪{∞}N∈N∪{∞}, P(|Q|>0)>0P(|Q|>0)>0 and nonnegative weights {Ci}{Ci}, and {Ri}i∈N{Ri}i∈N is a sequence of i.i.d. copies of RR independent of (Q,N,C1,C2,…)(Q,N,C1,C2,…); =D denotes equality in distribution. Furthermore, when Q>0Q>0 this recursion can be transformed into its additive equivalent, which corresponds to the maximum of a branching random walk and is also known as a high-order Lindley equation. We show that, under natural conditions, the asymptotic behavior of RR is power-law, i.e., P(|R|>x)∼Hx−αP(|R|>x)∼Hx−α, for some α>0α>0 and H>0H>0. This has direct implications for the tail behavior of other well known branching recursions.