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Maximums on trees

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  • Jelenković, Predrag R.
  • Olvera-Cravioto, Mariana

Abstract

We study the minimal/endogenous solution R to the maximum recursion on weighted branching trees given by R=D(⋁i=1NCiRi)∨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,…); =D 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.

Suggested Citation

  • Jelenković, Predrag R. & Olvera-Cravioto, Mariana, 2015. "Maximums on trees," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 217-232.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:1:p:217-232
    DOI: 10.1016/j.spa.2014.09.004
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    References listed on IDEAS

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    1. Biggins, J. D., 1998. "Lindley-type equations in the branching random walk," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 105-133, June.
    2. Buraczewski, Dariusz & Damek, Ewa & Mentemeier, Sebastian & Mirek, Mariusz, 2013. "Heavy tailed solutions of multivariate smoothing transforms," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1947-1986.
    3. Karpelevich, F. I. & Kelbert, M. Ya. & Suhov, Yu. M., 1994. "Higher-order Lindley equations," Stochastic Processes and their Applications, Elsevier, vol. 53(1), pages 65-96, September.
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    Cited by:

    1. Mariana Olvera-Cravioto & Octavio Ruiz-Lacedelli, 2021. "Stationary Waiting Time in Parallel Queues with Synchronization," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 1-27, February.

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