2014/05/24 by Predrag R. Jelenković, Jelenkovic, Predrag R., Mariana Olvera‐Cravioto +1
Mathematics · Physics and Astronomy · #60F10 #60H25 #60J80 #60K05 #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1405.6265
openalex publication_date 2014/05/24 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
We study the minimal/endogenous solution R to the maximum recursion on weighted branching trees given by R\stackrelD=(\bigveei=1NCiRi )\vee Q, where (Q,N,C1,C2,…) is a random vector with N∈ ℕ∪\∞\, P(|Q|>0)>0 and nonnegative weights \Ci\, and \Ri\i∈ℕ is a sequence of i.i.d. copies of R independent of (Q,N,C1,C2,…); \stackrelD= denotes equality in distribution. Furthermore, when Q>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 R is power-law, i.e., P(|R|>x)∼ Hx-α, for some α>0 and H>0. This has direct implications for the tail behavior of other well known branching recursions.