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 Bandyopadhyay and Ghosh  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 Gantert and Höfelsauer .
The author would like to thank the Council of Scientific and Industrial Research, Government of India and the Indian Statistical Institute, Kolkata for supporting his doctoral research.
This work is part of the author’s Ph.D. dissertation and the author wishes to thank Antar Bandyopadhyay for suggesting the problem and also for various discussions which he had with him as the Ph.D. supervisor. The author also thanks the anonymous referee, whose careful reading and detailed comments have helped to improve the paper.
"Large deviations for the right-most position of a last progeny modified branching random walk." Electron. Commun. Probab. 27 1 - 13, 2022. https://doi.org/10.1214/22-ECP446