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Large deviations for the right-most position of a last progeny modified branching random walk. (English) Zbl 1483.60043

Summary: In this work, we consider a modification of the usual Branching Random Walk (BRW), where we give certain independent and identically distributed (i.i.d.) displacements to all the particles at the \(n\)-th generation, which may be different from the driving increment distribution. This model was first introduced by A. Bandyopadhyay and P. P. Ghosh [“Right-most position of a last progeny modified branching random walk”, Preprint, arXiv:2106.02880] and they termed it as Last Progeny Modified Branching Random Walk (LPM-BRW). Under very minimal assumptions, we derive the large deviation principle (LDP) for the right-most position of a particle in generation \(n\). As a byproduct, we also complete the LDP for the classical model, which complements the earlier work by N. Gantert and T. Höfelsauer [Electron. Commun. Probab. 23, Paper No. 34, 12 p. (2018; Zbl 1394.60019)].

MSC:

60F10 Large deviations
60G50 Sums of independent random variables; random walks
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Citations:

Zbl 1394.60019
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References:

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