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 form $\frac 1 \theta \log\frac X E$, where $\theta$ is a positive parameter, $X$ has arbitrary distribution $\mu$ and $E$ is exponential with parameter 1, independent of $X$. Working under finite mean assumption for $\mu$, they proved almost sure convergence of the rightmost position to a constant limit, and identified the weak centered asymptotics when $\theta$ does not exceed certain critical parameter $\theta_0$. This paper complements their work by providing weak centered asymptotics for the case when $\theta > \theta_0$ and extending the results to $\mu$ 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 form $\alpha n + c \log n$, with constants $\alpha$, $c$ depending on the ratio of $\theta$ and $\theta_0$. We describe the limiting distribution and provide explicitly the constants appearing in the centering.