Branching random walks with regularly varying perturbations

We consider a modification of classical branching random walk, where we add i.i.d. perturbations to the positions of the particles in each generation. In this model, which was introduced and studied by Bandyopadhyay and Ghosh (2023), perturbations take the form 1/θ log X/E, where θ is a positive parameter, X has an arbitrary distribution μ supported on ℝ*+ and E is exponential with parameter 1, independent of X. Working under finite mean assumption for μ, they proved almost sure convergence of the rightmost position to a constant limit, and identified the weak centered asymptotics when θ does not exceed a certain critical parameter θ0. This paper complements their work by providing weak centered asymptotics for the case when θ > θ0 (this case was previously considered only in the case μ = δ0) and extending the results to μ with regularly varying tails. We prove almost sure convergence of the rightmost position and identify the appropriate centering for the weak convergence, which is of the form αn + c log n, with constants α, c depending on the ratio of θ and θ0. We describe the limiting distribution and provide explicitly the constants appearing in the centering.