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Lindley-type equations in the branching random walk

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  • Biggins, J. D.

Abstract

An analogue of the Lindley equation for random walk is studied in the context of the branching random walk, taking up the studies of Karpelevich, Kelbert and Suhov [(1993a) In: Boccara, N., Goles, E., Martinez, S., Picco, P. (Eds.), Cellular Automata and Cooperative Behaviour. Kluwer, Dordrecht, pp. 323-342; (1994a) Stochast. Process. Appl. 53, 65-96]. The main results are: (i) close to necessary conditions for the equation to have a solution, (ii) mild conditions for there to be a one-parameter family of solutions and (iii) mild conditions for this family to be the only possible solutions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:75:y:1998:i:1:p:105-133
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    References listed on IDEAS

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    1. 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. Iksanov, Alexander & Kolesko, Konrad & Meiners, Matthias, 2019. "Stable-like fluctuations of Biggins’ martingales," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4480-4499.
    2. Onno Boxma & Andreas Löpker & Michel Mandjes & Zbigniew Palmowski, 2021. "A multiplicative version of the Lindley recursion," Queueing Systems: Theory and Applications, Springer, vol. 98(3), pages 225-245, August.
    3. Fraiman, Nicolas & Lin, Tzu-Chi & Olvera-Cravioto, Mariana, 2023. "Stochastic recursions on directed random graphs," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    4. Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.
    5. 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.
    6. Jelenković, Predrag R. & Olvera-Cravioto, Mariana, 2015. "Maximums on trees," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 217-232.

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    2. Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.
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